Patent Publication Number: US-2020279416-A1

Title: Computing and Displaying Asymptotes and Removable Discontinuities

Description:
BACKGROUND 
     Understanding the domain of functions is an important part of a mathematics curriculum. Accordingly, the ability to compute and display asymptotes and removable discontinuities of generalized rational functions is a desirable feature for educational use of handheld devices such as graphing calculators as such functionality allows users to better understand and visualize the domain of such functions. 
     SUMMARY 
     Embodiments of the present invention relate to methods and systems for computing and displaying asymptotes and removable discontinuities. In one aspect, a method for evaluating a generalized rational function on a digital device is provided that includes computing, by at least one processor of the digital device, all discontinuities of the generalized rational function, determining, for each discontinuity, whether or not the discontinuity is a removable discontinuity, where the determining comprises determining, by the at least one processor, whether or not the generalized random function approaches a point near the discontinuity, and displaying each removable discontinuity on a display screen by the at least one processor. 
     In one aspect, a method for evaluating a generalized rational function on a handheld graphing calculator is provided that includes determining, by a processor of the handheld graphing calculator, whether or not the generalized rational function has at least one asymptote, and displaying, by the processor, the at least one asymptote on a display screen when the generalized rational function has the at least one asymptote. 
     In one aspect, a digital device is provided that includes a memory storing software instructions for evaluating a generalized rational function and at least one processor coupled to the memory to execute the software instructions. The software instructions include software instructions to compute all discontinuities of the generalized rational function, determine, for each discontinuity, whether or not the discontinuity is a removable discontinuity, wherein the determining comprises determining whether or not the generalized random function approaches a point near the discontinuity, and display each removable discontinuity on a display screen by the at least one processor. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an example of a graph of a rational function; 
         FIG. 2  is a flow diagram of a method for determining horizontal and oblique asymptotes of a function; 
         FIG. 3  is a flow diagram of a method for determining the vertical asymptotes of a generalized rational function; 
         FIG. 4  is a flow diagram of a method for determining the removable discontinuities of a generalized rational function; 
         FIG. 5  is an example illustrating determining discontinuities of a generalized rational function; 
         FIGS. 6-14  are examples of user interfaces; and 
         FIGS. 15 and 16  are block diagrams of two example digital devices. 
     
    
    
     DETAILED DESCRIPTION 
     Specific embodiments of the disclosure will now be described in detail with reference to the accompanying figures. Like elements in the various figures are denoted by like reference numerals for consistency. 
     Embodiments of the disclosure provide for computing and displaying asymptotes and removable discontinuities of generalized rational functions on digital devices such as, for example, handheld graphing calculators, tablets, etc. Having the ability to calculate and graph asymptotes and removable discontinuities can improve understanding of rational function limits and domains, e.g., for students in a classroom, by providing more accurate visual displays. For example, if students are learning about domains of functions, the ability to calculate and graph removable discontinuities and vertical asymptotes on a digital device can improve understanding of types of discontinuities. Further, if students are learning about limits of functions, the ability to calculate and graph asymptotes can improve the understanding of the subject matter. In addition, for all users, the ability to calculate and graphically display removable discontinuities of functions provides a more accurate representation of the functions. Further, the computation of asymptotes and removable discontinuities are as accurate as numerical results and can be exact results in some embodiments. 
     A generalized rational function can be a rational function or a polynomial raised to a negative integer power. Further, a generalized rational function divided by a polynomial is also a generalized rational function. The sum, difference, and/or product of generalized functions are also generalized rational functions. In addition, a generalized rational function raised to a nonnegative integer power is a generalized rational function. A rational function is a function expressed as a ratio of polynomials, i.e., an algebraic fraction in which both the numerator and denominator are polynomials. For example, the function  100  of  FIG. 1 , which includes the sum and difference of four rational functions, is a generalized rational function. Further, a generalized rational function can be reduced to a rational function with appropriate domain restrictions. For example, the generalized rational function  100  of  FIG. 1  equals the rational function (2x 2 −10x+3)/(x−5) except where x=0 and x=6. 
     The domain of a function f(x) is the set of all values of x for which the function is defined and the range of the function is the set of all values that f(x) takes. For a generalized rational function, the domain is all real numbers except those numbers that result in zeros in the denominator of any division operation in the generalized rational function. Further, the graph of a generalized rational function has a discontinuity at each value excluded from the domain. These discontinuities can be one of two types: removable discontinuities, also referred to as holes, and infinite discontinuities, also referred to as vertical asymptotes. A generalized rational function f(x) has a removable discontinuity at the value a if the discontinuity at a can be removed by changing the value of the function at a to a finite value i.e., making 
     
       
         
           
             
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     if the limit is finite. If the function becomes infinitely large or infinitely small as x converges to a finite value, e.g., a, from either direction, then a vertical asymptote exists at that finite value of x, i.e., at x=a. 
     Holes or removable discontinuities are the domain restrictions that are removed when a generalized rational function is reduced to a rational function in its simplest form. The example generalized rational function  100  of  FIG. 1  can be reduced to (2x 2 −10x+3)/(x−5). The domain restrictions that are removed by this reduction occur at are x=0 and x=6. Therefore, the generalized rational function  100  has a vertical asymptote at x=5 and has two removable discontinuities with x coordinates of 0 and 6. The y coordinates of the removable discontinuities can be calculated by substituting the x coordinate in the reduced function. Thus, the coordinates of the removable discontinuities are (0, −3/5) and (6, 15). The graph  102  illustrates the two removable discontinuities  104 ,  106  and the vertical asymptote  108  of the generalized rational function  100 . 
     Generalized rational functions may also have other types of asymptotes including horizontal asymptotes and oblique asymptotes. A horizontal asymptote means the output of the generalized rational function approaches a constant value as the value x approaches ∞ or −∞. An oblique asymptote only occurs when the numerator of the reduced rational function from the generalized rational function has a degree that is one higher than the degree of the denominator. For example, the generalized rational function  100  of  FIG. 1  meets this criterion and the graph  102  shows the resulting oblique asymptote  110 . 
       FIG. 2  is a flow diagram of a method for determining horizontal and oblique asymptotes of a function f(x), e.g., a generalized rational function, suitable for execution on a digital device. Initially, a determination is made as to whether or not the graph of the function f(x) includes a horizontal or oblique asymptote. In general, the graph of the function f(x) includes such an asymptote if the graph becomes linear as x approaches ∞ or −∞. Accordingly, the slope of the graph as x approaches ∞ or −∞, i.e., 
     
       
         
           
             
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     is computed  200  and the magnitude of m is compared to ∞  202 . If m is defined, which is always true of generalized rational functions, and finite, i.e., the magnitude of m is less than ∞  202 , there is either a horizontal asymptote or an oblique asymptote. Otherwise, there is no horizontal or oblique asymptote. 
     For example, if f(x)=x 2 , then 
     
       
         
           
             
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     the graph does not behave as a line as the value of x grows so the graph has no horizontal or oblique asymptote. However, if f(x)=(5x 2 −1)/(4x), then 
     
       
         
           
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     In this case, the graph behaves like a line as x grows, and more specifically, the graph behaves like a line with slope 5/4. 
     If the function has a horizontal or oblique asymptote  202 , then the y-intercept of the line is computed  204  as per 
     
       
         
           
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     If the y-intercept is finite, i.e., the magnitude of a is less than ∞  206 , there is either a horizontal or oblique asymptote. Otherwise, there is no asymptote. Note that for a generalized rational function, the magnitude of the y-intercept is always less than infinity. Accordingly, the step of checking the value of the y-intercept a is optional and can be excluded if the function f(x) is a generalized rational function. 
     Given the y-intercept a and the slope m, the asymptote of the function f(x) can be defined as per y=m*x+a. If m=0 208, i.e., the slope of the line is 0, then there is a horizontal asymptote at y=a and the horizontal asymptote is displayed  210  on the display screen of the digital device. Otherwise, there is an oblique asymptote defined by y=m*x+a and the oblique asymptote is displayed  212  on the display screen of the digital device. Continuing the previous example, given f(x)=(5x 2 −1)/(4x) and m=5/4, 
     
       
         
           
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     Thus, the equation of the oblique asymptote of f(x) is y=5/4*x+0=5/4*x. 
     As is explained in more detail herein, the horizontal or oblique asymptote can be displayed, for example, as part of the graph of the function f(x) or by textual display. In some embodiments, when only the horizontal asymptote is to be determined, the method begins at step  204  with m=0 as the slope of a horizontal asymptote is always zero. 
       FIG. 3  is a flow diagram of a method for determining the vertical asymptotes of a generalized rational function f(x) suitable for execution on a digital device. Initially, the generalized rational function f(x) is reduced to a simplified rational function to hide any removable discontinuities  300 , i.e., all factors common to both the numerator and denominator are cancelled out and all terms are combined. This step is performed to simplify the calculations performed in subsequent steps to enhance performance and can be eliminated in other embodiments. The discontinuities of the resulting generalized rational function g(x) are then determined  302 . For example, if 
     
       
         
           
             
               
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     then 
     
       
         
           
             
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     and there is a discontinuity at x=0. Any suitable technique for determining the discontinuities can be used. In some embodiments, the method for determining the discontinuities as described herein in reference to Table 1 and  FIG. 5  is used. 
     If no discontinuities are found  303 , the generalized rational function f(x) has no vertical asymptotes. If one or more discontinuities are found  303 , then each discontinuity as is checked  304 - 308  to determine if it is a vertical asymptote. There is a vertical asymptote at a discontinuity as if the graph of g(x) approaches ∞ or −∞ as the value of x approaches as from any direction. More specifically, the limit of g(x) as the value of x approaches the discontinuity a i  from the left direction, i.e., 
     
       
         
           
             
               
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     is calculated. If the limit is equal to ±∞, then there is a vertical asymptote. Otherwise, the limit of g(x) as the value of x approaches the discontinuity a from the right direction, i.e., 
     
       
         
           
             
               
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     is calculated and compared to ±∞. If either value is equal to +∞  304 , then there is a vertical asymptote at x=a i  which is displayed  306  on the display screen of the digital device. Otherwise, there is no vertical asymptote at the discontinuity a i . As is explained in more detail herein, the vertical asymptote(s) can be displayed, for example, as part of the graph of the function f(x) or by textual display. Continuing the previous example, for 
     
       
         
           
             
               
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     so there is a vertical asymptote at x=a 0 , where a 0 =0. 
       FIG. 4  is a flow diagram of a method for determining the removable discontinuities of a generalized rational function f(x) suitable for execution on a digital device. Initially, the discontinuities of the generalized rational function f(x) are determined  400 . Any suitable technique for determining the discontinuities can be used. In some embodiments, the method for determining the discontinuities as described herein in reference to Table 1 and  FIG. 5  is used. If no discontinuities are found  401 , the generalized rational function f(x) has no removable discontinuities. If one or more discontinuities are found  401 , then each discontinuity a is checked  402 - 408  to determine if it is a removable discontinuity. 
     Unlike asymptotes, removable discontinuities are points and as such, have an x-coordinate and a y-coordinate. If a function has a removable discontinuity, the function can be made continuous at the x-coordinate location of the removable discontinuity by setting the function to the corresponding y-coordinate, i.e., by filling the hole with a point. Accordingly, to determine if a discontinuity as is a removable discontinuity, the behavior of the function, i.e., the y-coordinate values, as the function approaches the discontinuity as from both directions are examined. To determine the behavior, 
     
       
         
           
             
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     is computed  402 . If b i  is defined, i.e., the behavior of the function is known and the same from the left and right, and is finite, i.e., the magnitude of b i  is less than ∞  404 , then b i  is a single finite value which indicates that the function approaches a point at the discontinuity a and the removable discontinuity at (a i , b i ) is displayed  406 . Otherwise, the discontinuity a is not a removable discontinuity. As is explained in more detail herein, the removable discontinuities can be displayed, for example, as part of the graph of the function f(x) or by displaying the coordinates of the removable discontinuities textually. 
     Consider the following three examples. If f(x)=(x 2 −1)/x, there is a discontinuity at x=0 and the function behaves differently as the discontinuity is approached from the two different directions: 
     
       
         
           
             
               
                 
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     As a result, b i  is not defined (step  404 ) and the discontinuity is not removable. 
     If f(x)=(x 3 −1)/x 2 , there is a discontinuity at x=0 and the function behaves the same as the discontinuity is approached from the two different directions: 
     
       
         
           
             
               
                 
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     As a result, b 0  is defined, i.e. 
     
       
         
           
             
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               0 
             
             = 
             
               
                 
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                     → 
                     
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                       0 
                     
                   
                 
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                         x 
                         3 
                       
                       - 
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                       2 
                     
                   
                 
                 = 
                 
                   - 
                   
                     ∞ 
                     . 
                   
                 
               
             
           
         
       
     
     However, b 0  is not finite which means the function cannot be made continuous by replacing the discontinuity with a point. Accordingly, the discontinuity is not removable. 
     If f(x)=(x 2 −x)/x, there is a discontinuity at x=0 the function behaves the same as the discontinuity is approached from the two different directions: 
     
       
         
           
             
               
                 
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                     x 
                     → 
                     
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                       0 
                       - 
                     
                   
                 
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                     ( 
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                     ) 
                   
                 
               
               = 
               
                 
                   
                     lim 
                     
                       x 
                       → 
                       
                         0 
                         - 
                       
                     
                   
                    
                   
                     
                       
                         x 
                         2 
                       
                       - 
                       x 
                     
                     x 
                   
                 
                 = 
                 
                   
                     
                       lim 
                       
                         x 
                         → 
                         
                           0 
                           - 
                         
                       
                     
                      
                     
                       ( 
                       
                         x 
                         - 
                         1 
                       
                       ) 
                     
                   
                   = 
                   
                     - 
                     1 
                   
                 
               
             
              
             
               
 
             
              
             
               
                 
                   lim 
                   
                     x 
                     → 
                     
                       a 
                       0 
                       + 
                     
                   
                 
                  
                 
                   f 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               = 
               
                 
                   
                     lim 
                     
                       x 
                       → 
                       
                         0 
                         + 
                       
                     
                   
                    
                   
                     
                       
                         x 
                         2 
                       
                       - 
                       x 
                     
                     x 
                   
                 
                 = 
                 
                   
                     
                       lim 
                       
                         x 
                         → 
                         
                           0 
                           + 
                         
                       
                     
                      
                     
                       ( 
                       
                         x 
                         - 
                         1 
                       
                       ) 
                     
                   
                   = 
                   
                     - 
                     1 
                   
                 
               
             
           
         
       
     
     As a result, b 0  is defined, i.e.: 
     
       
         
           
             
               b 
               0 
             
             = 
             
               
                 
                   lim 
                   
                     x 
                     → 
                     
                       a 
                       0 
                     
                   
                 
                  
                 
                   f 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               = 
               
                 
                   
                     lim 
                     
                       x 
                       → 
                       0 
                     
                   
                    
                   
                     
                       
                         x 
                         2 
                       
                       - 
                       x 
                     
                     χ 
                   
                 
                 = 
                 
                   - 
                   
                     1 
                     . 
                   
                 
               
             
           
         
       
     
     Furthermore, b 0  is finite which means that the function can be made locally continuous by replacing the discontinuity with a point. Thus, there is a removable discontinuity at (0,−1). 
     Table 1 is pseudo code of a method for determining all discontinuities in a generalized rational function f(x). In general, the pseudo code walks the function and recursively examines the domain to locate the discontinuities, i.e., the values of x that cause the denominators in the generalized rational function to become zero. More specifically, the pseudo code parses a generalized rational function from left to right, recursing on each operand of the top level operators negation, plus, minus, and multiplication identified in a recursion iteration. When the top level operator is division, the method recurses on the numerator of the division operation and determines the zeros, also referred to as roots, of the denominator. A zero is an input value that produces an output of zero. A zero or root of a denominator is a discontinuity of the input generalized rational function. 
     If the top level operation in a recursion iteration is exponentiation and the exponent is positive, the method recurses on the base. If the top level operation is exponentiation and the exponent is negative, the zeros of the base are determined. A zero in the base of such an exponentiation operation is a discontinuity of the input generalized rational function. If the input to a recursion iteration is a constant or a variable, i.e., there is no top level operation, the input is continuous and the result of the iteration is no discontinuities. The output of the pseudo code is a list of the identified discontinuities referred to as list_of_discontinuities. 
     
       
         
           
               
             
               
                 TABLE 1 
               
               
                   
               
             
            
               
                 calculate_discontinuities (generalized rational function) 
               
               
                 {if (generalized rational function is constant or a variable) 
               
            
           
           
               
               
            
               
                   
                 {// The generalized rational function is a constant, such as 2 or sqrt(3) 
               
            
           
           
               
            
               
                 or pi, or just x and has no discontinuities. 
               
            
           
           
               
               
            
               
                   
                 list_of_discontinuities = { };} 
               
            
           
           
               
            
               
                  else if (top level operator is negation sign) 
               
            
           
           
               
               
            
               
                   
                 {// The generalized rational function is the negative of a function. 
               
            
           
           
               
            
               
                 Calculate the discontinuities of the function being negated 
               
            
           
           
               
               
            
               
                   
                 list_of_discontinuities = calculate_discontinuities (operand);} 
               
            
           
           
               
            
               
                 else if (top level operator is addition, subtraction, or multiplication) 
               
            
           
           
               
               
            
               
                   
                 {// The generalized rational function is sum, difference, and/or 
               
            
           
           
               
            
               
                 product of two generalized rational functions. 
               
            
           
           
               
               
            
               
                   
                  // Calculate the discontinuities of left operand and right operand. 
               
            
           
           
               
               
            
               
                   
                 list_of_discontinuities_1 = calculate_discontinuities (left 
               
               
                   
                 operand); 
               
               
                   
                 list_of_discontinuities_2 = calculate_discontinuities (right 
               
               
                   
                 operand); 
               
               
                   
                 list_of_discontinuities = union of list_of_discontinuites_1 
               
               
                   
                 and 
               
            
           
           
               
            
               
                 list_of_discontinuities_2 } 
               
               
                  else if (top level operator is division) 
               
            
           
           
               
               
            
               
                   
                 {// The generalized rational function is the quotient of two functions 
               
            
           
           
               
               
            
               
                   
                 If (denominator is a polynomial) 
               
               
                   
                 { list_of_discontinuities_1 = calculate_discontinuities 
               
               
                   
                 (numerator); 
               
               
                   
                  list_of_discontinuities_2 = zeros of denominator; 
               
               
                   
                  list_of_discontinuities = union of list_of_discontinuites_1 
               
               
                   
                  and 
               
            
           
           
               
            
               
                 list_of_discontinuites_2; } 
               
            
           
           
               
               
            
               
                   
                 else 
               
               
                   
                 {//The input is not a generalized rational function 
               
            
           
           
               
               
            
               
                   
                 Throw an error;}} 
               
            
           
           
               
               
            
               
                   
                 else if (top level operator is exponentiation) 
               
               
                   
                 {// The generalized rational function is base power   
               
            
           
           
               
               
            
               
                   
                 If (power is integer) 
               
               
                   
                 {if (power is negative) 
               
            
           
           
               
               
            
               
                   
                 {if (base is polynomial) 
               
            
           
           
               
               
            
               
                   
                 {// For example, the generalized rational function may 
               
               
                   
                 be (x + 1) −2   
               
            
           
           
               
               
            
               
                   
                 list_of_discontinuities = zeros of base;} 
               
            
           
           
               
               
            
               
                   
                 else 
               
               
                   
                 {// The input is not a generalized rational function 
               
            
           
           
               
               
            
               
                   
                 Throw an error; } } 
               
            
           
           
               
               
            
               
                   
                 else 
               
               
                   
                 {// For example, the rational function may be (1/(x−1)) 3   
               
            
           
           
               
               
            
               
                   
                 list_of_discontinuities = calculate_discontinuities 
               
               
                   
                 (base);}} 
               
            
           
           
               
               
            
               
                   
                 else 
               
               
                   
                 {// The input is not a generalized rational function 
               
            
           
           
               
               
            
               
                   
                 Throw an error; } } 
               
            
           
           
               
               
            
               
                   
                 else 
               
               
                   
                 {// The input is not a generalized rational function 
               
            
           
           
               
               
            
               
                   
                 Throw an error; } 
               
            
           
           
               
               
            
               
                   
                 return list_of_discontinuities;} 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 5  is an example recursion tree illustrating the operation of the pseudo code of Table 1. The root node  500  of the recursion tree shows the input to the pseudo code which is the generalized rational function 
     
       
         
           
             
               
                 
                   x 
                    
                   
                     1 
                     x 
                   
                 
                 
                   x 
                   + 
                   1 
                 
               
               + 
               
                 1 
                 
                   x 
                   - 
                   1 
                 
               
             
             . 
           
         
       
     
     In the first recursion iteration, the top level operator is addition. The pseudo code recurses on the left operand 
     
       
         
           
             
               x 
               - 
               
                 1 
                 x 
               
             
             
               x 
               + 
               1 
             
           
         
       
     
     and the right operand 
     
       
         
           
             1 
             
               x 
               - 
               1 
             
           
         
       
     
     of the addition, as indicated by the respective child nodes  502 ,  504 . The recursive processing continues at the left child node  502  where the top level operator is division. The pseudo code recurses on the numerator 
     
       
         
           
             x 
             - 
             
               1 
               x 
             
           
         
       
     
     and determines the zeros of the denominator x+1, as indicated by the respective child nodes  508 ,  510 . 
     The recursive processing continues at the left child node  508  where the top level operator is subtraction. The pseudo code recurses on the left operand x and the right operand 
     
       
         
           
             1 
             x 
           
         
       
     
     of the subtraction, as indicated by the respective child nodes  512 ,  514 . The recursive processing continues at the left child node  512 , which is the variable x. The variable x is continuous, so there is no discontinuity. The recursive processing then backtracks to the node  508  and moves to the right child node  514 . At the right child node  514 , the top level operation is division. The pseudo code recurses on the numerator  1  and determines the zeros of the denominator x, as indicated by the respective child nodes  516 ,  518 . 
     The recursive processing continues at the left child node  516 , which is the constant  1 . The constant  1  is continuous, so there is no discontinuity. The recursive processing then backtracks to the node  514  and moves to the right child node  518  to determine the zeros of the denominator x. The denominator is simply the variable x, so the single zero of the denominator is 0, which will be added to the final list of discontinuities output by the pseudo code as the recursive processing unwinds. The recursive processing then backtracks to the node  514 , then to the node  508 , and ultimately to the node  502 . From the node  502 , the recursive processing moves to the right child node  510  to determine the zeros of the denominator x+1. The single zero of the denominator is −1, which will be added to the final list of discontinuities output by the pseudo code as the recursive processing unwinds. At this point, two discontinuities have been identified, x=0 and x=−1. 
     The recursive processing then backtracks to the node  502  and ultimately to the node  500 . From the node  500 , the recursive processing moves to the right child node  504 . At the right child node  504 , the top level operator is division. The pseudo code recurses on the numerator  1  and determines the zeros of the denominator x−1, as indicated by the respective child nodes  520 ,  522 . The recursive processing continues at the left child node  520 , which is the constant  1 . The constant  1  is continuous, so there is no discontinuity. The recursive processing then backtracks to the node  504  and moves to the right child node  522  to determine the zeros of the denominator x−1. The single zero of the denominator is 1, which will be added to the final list of discontinuities output by the pseudo code as the recursive processing unwinds. At this point, three discontinuities have been identified, x=0, x=−1, and x=1. The recursive processing backtracks to the node  504  and ultimately to the node  500 , where processing terminates and the list of discontinuities {0, −1, 1} is returned. 
       FIGS. 6-14  are examples of various interfaces to the above described methods. Each example shows two example interfaces, a graphical interface on the right and a textual interface on the left, as such interfaces might appear on the display screen of a digital device, e.g., a handheld graphing calculator. The example of  FIG. 6  illustrates determining all the asymptotes of a function. In the textual interface  600 , the user enters  604  the name of the command, in this case “asymptote”, and the function. The command asymptote causes implementations of the above methods of  FIG. 2  and  FIG. 3  for determining horizontal, vertical, and oblique asymptotes to be executed to determine the asymptotes of the function. The result  606  of the command, which in this example is a vertical asymptote at x=2 and a horizontal asymptote at y=1, is displayed textually. 
     An alternative way of finding all the asymptotes of a function is to display the graph of the function on the display screen and allow the user to select an option for showing all asymptotes from a menu activated by clicking on the graph. The selection of the option causes implementations of the above methods of  FIG. 2  and  FIG. 3  for determining horizontal, vertical, and oblique asymptotes to be executed to determine the asymptotes of the graphed function and displays the results graphically. The graphical interface  602  shows the results of selecting such an option for the graph of the same function illustrated in the textual interface  600 . 
       FIG. 7  is another example of determining all asymptotes of a function. In the textual interface  700 , the user enters  704  the name of the command and the function. The command asymptote causes implementations of the above methods of  FIG. 2  and  FIG. 3  for determining horizontal, vertical, and oblique asymptotes to be executed to determine the asymptotes of the function. The result  706  of the command, which in this example is a vertical asymptote at x=1 and an oblique asymptote at y=x+2, is displayed textually. The graphical interface  702  shows the results of selecting an option to show all asymptotes from a menu activated from the graph of the same function illustrated in the textual interface  700 . 
     The example of  FIG. 8  illustrates determining the horizontal asymptote of a function. In the textual interface  800 , the user enters  804  the name of the command, in this case “hasymptote”, and the function. The command hasymptote causes an implementation of the above method of  FIG. 2  for determining oblique and horizontal asymptotes to be executed to determine the horizontal asymptote of the function. As previously noted, if only the horizontal asymptote is sought, the method can begin at step  204  with m=0. The result  806  of the command, which in this example is a horizontal asymptote at y=1, is displayed textually. 
     An alternative way of finding the horizontal asymptote of a function is to display the graph of the function on the display screen and allow the user to select an option for showing the horizontal asymptote from a menu activated by clicking on the graph. The selection of the option causes an implementation of the above method of  FIG. 2  for determining horizontal and oblique asymptotes to be executed to determine the horizontal asymptote of the graphed function and displays the results graphically. The graphical interface  802  shows the results of selecting such an option for the graph of the same function illustrated in the textual interface  800 . 
       FIG. 9  is another example of determining the horizontal asymptote of a function. In the textual interface  900 , the user enters  904  the name of the command and the function. The command “hasymptote” causes an implementation of the above method of  FIG. 2  for determining horizontal and oblique asymptotes to be executed to determine the horizontal asymptote of the function. The result  906  of the command, which is false, is displayed textually. The graphical interface  902  shows the results of selecting an option to show the horizontal asymptote from a menu activated from the graph of the same function illustrated in the textual interface  900 . 
     The example of  FIG. 10  illustrates determining the vertical asymptotes of a function. In the textual interface  1000 , the user enters  1004  the name of the command, in this case “vasymptote”, and the function. The command vasymptote causes an implementation of the above method of  FIG. 3  for determining vertical asymptotes to be executed to determine the vertical asymptotes of the function. The result  1006  of the command, which in this example is a vertical asymptote at x=2, is displayed textually. 
     An alternative way of finding the vertical asymptotes of a function is to display the graph of the function on the display screen and allow the user to select an option for showing the vertical asymptotes from a menu activated by clicking on the graph. The selection of the option causes an implementation of the above method of  FIG. 3  for determining vertical asymptotes to be executed to determine the vertical asymptotes of the graphed function and displays the results graphically. The graphical interface  1002  shows the results of selecting such an option for the graph of the same function illustrated in the textual interface  1000 . 
       FIG. 11  is another example of determining the vertical asymptotes of a function. In the textual interface  1100 , the user enters  1104  the name of the command and the function. The command “vasymptote” causes an implementation of the above method of  FIG. 3  for determining vertical asymptotes to be executed to determine the vertical asymptotes of the function. The result  1106  of the command, which is a vertical asymptote at x=1, is displayed textually. The graphical interface  1102  shows the results of selecting an option to show the vertical asymptotes from a menu activated from the graph of the same function illustrated in the textual interface  1100 . 
     The example of  FIG. 12  illustrates determining the oblique asymptote of a function. In the textual interface  1200 , the user enters  1204  the name of the command, in this case “oasymptote”, and the function. The command oasymptote causes an implementation of the above method of  FIG. 2  for determining horizontal and oblique asymptotes to be executed to determine the oblique asymptote of the function. The result  1206  of the command, which in this example is false, is displayed textually. 
     An alternative way of finding the oblique asymptote of a function is to display the graph of the function on the display screen and allow the user to select an option for showing the oblique asymptote from a menu activated by clicking on the graph. The selection of the option causes an implementation of the above method of  FIG. 2  for determining horizontal and oblique asymptotes to be executed to determine the oblique asymptote of the graphed function and displays the results graphically. The graphical interface  1202  shows the results of selecting such an option for the graph of the same function illustrated in the textual interface  1200 . 
       FIG. 13  is another example of determining the oblique asymptote of a function. In the textual interface  1300 , the user enters  1304  the name of the command and the function. The command “oasymptote” causes an implementation of the above method of  FIG. 2  for determining horizontal and oblique asymptotes to be executed to determine the oblique asymptote of the function. The result  1306  of the command, which is an oblique asymptote at y=x+2, is displayed textually. The graphical interface  1302  shows the results of selecting an option to show the oblique asymptote from a menu activated from the graph of the same function illustrated in the textual interface  1300 . 
     The example of  FIG. 14  illustrates determining the holes (removable discontinuities) of a function. In the textual interface  1400 , the user enters  1404  the name of the command, in this case “holes”, and the function. The command holes causes an implementation of the above method of  FIG. 4  for determining removable discontinuities to be executed to determine the removable discontinuities of the function. The result  1406  of the command, which in this example is that there is a removable discontinuity at x=1 and y=1, is displayed textually. 
     An alternative way of finding the removable discontinuities of a function is to display the graph of the function on the display screen and allow the user to select an option for showing the removable discontinuities from a menu activated by clicking on the graph. The selection of the option causes an implementation of the above method of  FIG. 4  for determining removable discontinuities to be executed to determine the removable discontinuities of the graphed function and displays the results graphically. The graphical interface  1402  shows the results of selecting such an option for the graph of the same function illustrated in the textual interface  1400 . 
     Embodiments of the methods described herein can be implemented on any suitably configured digital device, e.g., a handheld graphing calculator, a smart phone, a tablet, a laptop, or a computer system.  FIGS. 15 and 16  are diagrams of two example digital devices. 
       FIG. 15  is an example of a handheld graphing calculator  1500  configured to perform methods for determining asymptotes and removable discontinuities as described herein described herein. The handheld calculator  1500  includes a display screen  1504 , and a keypad  1502  that includes a touchpad  1506 . The display screen  1504  can be used to display, among other things, information input to applications executing on the handheld graphing calculator  1500  and various outputs of the applications. For example, the display screen  1504  may be used to display input of commands that activate the use of the methods described herein and the outputs of those commands, and/or the graphs of functions and the results of selecting options from menus that activate the use of the methods and the outputs of the selected options. The display screen  1504  may be, for example, an LCD display. 
     The keypad  1502  allows a user to enter data and functions and to start and interact with applications executing on the handheld graphing calculator  1500 . The keypad  1502  also includes an alphabetic keyboard for entering text. The touchpad  1506  allows a user to interact with the display  1504  by translating the motion and position of the user&#39;s fingers on the touchpad  1506  to provide functionality similar to using an external pointing device, e.g., a mouse. A user may use the touchpad  1506  to perform operations similar to using a pointing device on a computer system, e.g., scrolling the display  1504  content, pointer positioning, selecting, highlighting, etc. 
     The handheld graphing calculator  1500  includes a processor  1501  coupled to a memory unit  1512 , e.g., a non-transitory computer-readable medium, which may include one or both of memory for program storage, e.g., read-only memory (ROM), and memory for non-persistent data and program storage, e.g., random-access memory (RAM). In some embodiments, the program storage memory stores software programs and the memory for non-persistent data stores intermediate data and operating results. An input/output port  1508  provides connectivity to external devices, e.g., a wireless adaptor or wireless cradle. In one or more embodiments, the input/output port  1508  is a bi-directional connection such as a mini-A USB port. Also included in the handheld graphing calculator  1500  is an I/O interface  1510 . The I/O interface  1510  provides an interface to couple input devices such as the touchpad  1506  and the keypad  1502  to the processor  1501 . In some embodiments, the handheld calculator  1500  may also include an integrated wireless interface (not shown) or a port for connecting an external wireless interface (not shown). 
     In one or more embodiments, the memory unit  1512  stores software instructions to be executed by the processor  1501  to perform embodiments of the methods described herein. Further, in some such embodiments, the memory unit  1512  stores software instructions of a computer algebra system (CAS) that includes functionality to process equations and functions symbolically and functionality for graphing equations and functions. The CAS includes functionality to, for example, simplify rational functions, factor polynomials, compute limits of functions, compute zeros of functions, etc. One example of such a CAS is the CAS installed on the TI-Nspire™ series of graphing calculators available from Texas Instruments, Inc. The CAS is described in “TI-Nspire™ CAS Reference Guide,” Texas Instruments, Inc., 2006-2012, which is incorporated by reference herein in its entirety. 
       FIG. 16  is an example of a computer system  1600  configured to perform methods for determining asymptotes and removable discontinuities as described herein. The computer system  1600  includes a processing unit  1630  coupled to one or more input devices  1604  (e.g., a mouse, a keyboard, or the like), and one or more output devices, such as a display screen  1608 . In some embodiments, the display screen  1608  may be touch screen, thus allowing the display screen  1608  to also function as an input device. The processing unit  1630  may be, for example, a desktop computer, a workstation, a laptop computer, a tablet, a dedicated unit customized for a particular application, or the like. The display screen  1608  may be any suitable visual display unit such as, for example, a computer monitor, an LED, LCD, or plasma display, a television, a high definition television, or a combination thereof. The display screen  1608  can be used, for example, to display input of commands that activate the use of the methods described herein and the outputs of those commands, and/or the graphs of functions and the results of selecting options from menus that activate the use of the methods and the outputs of the selected options 
     The processing unit  1630  includes a processor  1618 , memory  1614 , a storage device  1616 , a video adapter  1612 , and an I/O interface  1610  connected by a bus. The bus may be one or more of any type of several bus architectures including a memory bus or memory controller, a peripheral bus, video bus, or the like. The processor  1618  may be any type of electronic data processor. For example, the processor  1618  may be a processor from Intel Corp., a processor from Advanced Micro Devices, Inc., a Reduced Instruction Set Computer (RISC), an Application-Specific Integrated Circuit (ASIC), or the like. The memory  1614 , e.g., a non-transitory computer-readable medium, can be any type of system memory such as static random access memory (SRAM), dynamic random access memory (DRAM), synchronous DRAM (SDRAM), read-only memory (ROM), a combination thereof, or the like. Further, the memory  1614  can include ROM for use at boot-up, and DRAM for data storage for use while executing programs. 
     The storage device  1616 , e.g., a non-transitory computer-readable medium, can include any type of storage device configured to store data, programs, and other information and to make the data, programs, and other information accessible via the bus. In one or more embodiments, the storage device  1616  stores software instructions to be executed by the processor  1618  to perform embodiments of the methods described herein. The storage device  1616  may be, for example, one or more of a hard disk drive, a magnetic disk drive, an optical disk drive, or the like. 
     The video adapter  1612  and the I/O interface  1610  provide interfaces to couple external input and output devices to the processing unit  1630 . The processing unit  1630  also includes a network interface  1624 . The network interface  1624  allows the processing unit  1630  to communicate with remote units via a network (not shown). The network interface  1624  may provide an interface for a wired link, such as an Ethernet cable or the like, or a wireless link. The computer system  1600  may also include other components not specifically shown. For example, the computer system  1600  may include power supplies, cables, a motherboard, removable storage media, cases, and the like. 
     Other Embodiments 
     While the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments can be devised which do not depart from the scope of the invention as disclosed herein. 
     For example, embodiments have been described herein in which a user enters a command to cause removable discontinuities for a generalized rational function to be computed and displayed. In some embodiments, any removable discontinuities of a generalized rational function are displayed by default. That is, when a user requests that generalized rational function be graphed, the removable discontinuities of the function are computed and displayed on the graph of the function automatically. Further, in some such embodiments, the automatic computation and display of removable discontinuities can be disabled and reenabled, e.g., when a student is using the digital device for an examination. Some examples of enabling and disabling functionality on a digital device that can be used are described in U.S. Pat. No. 8,499,014 and United States Patent Application Publication 2017/0011239, both of which are incorporated by reference herein. An example of an interface for selective enabling and disabling of functionality on the TI-Nspire™ series of graphing calculators for testing purposes is described in “TI-Nspire™ handhelds: Test preparation: Press-to-Test”, Texas Instruments, Inc., 2014. In addition, in embodiments having a CAS, the computation and display of removable discontinuities can be disabled/enabled by enabling and disabling the CAS. 
     In another example, embodiments have been described herein in which a user enters commands to cause asymptotes for a generalized rational function to be computed and displayed, either all asymptotes or selected types of asymptotes, e.g., vertical, horizontal, or oblique. In some embodiments, all asymptotes of a generalized rational function are displayed by default. That is, when a user requests that generalized rational function be graphed, the vertical, horizontal, and oblique asymptotes of the function are computed and displayed on the graph of the function automatically. Further, in some such embodiments, the automatic computation and display of asymptotes can be disabled and reenabled, e.g., when a student is using the digital device for an examination. Some examples of enabling and disabling functionality on a digital device that could be used are previously described herein. 
     In another example, embodiments are described herein in which a user enters textual commands to cause the computation and display of asymptotes and removable discontinuities. In some embodiments, the commands can be selected from menus, e.g., popup or drop down menus. 
     Software instructions implementing all or portions of methods described herein may be initially stored in a non-transitory computer-readable medium and loaded and executed by one or more processors. In some cases, the software instructions may be distributed via removable non-transitory computer-readable media, via a transmission path from non-transitory computer-readable media on another digital system, etc. Examples of non-transitory computer-readable media include non-writable storage media such as read-only memory devices, writable storage media such as disks, flash memory, memory, or a combination thereof. 
     It is therefore contemplated that the appended claims will cover any such modifications of the embodiments as fall within the true scope of the disclosure.