Limit of a function

In calculus, a branch of mathematics, the limit of a function is the behavior of a certain function near a selected input value for that function. Limits are one of the main calculus topics, along with derivatives,  integration, and differential equations.

 Definition of the limit 

The definition of the limit is as follows:
If the function  approaches a number  as  approaches a number , then 

The notation for the limit above is read as "The limit of  as  approaches  is ", or alternatively,  as   (reads " tends to  as  tends to "). Informally, this means that we can make  as close to  as possible—by making  sufficiently close to  from both sides (without making  equal to ).

Imagine we have a function such as . When ,   is undefined, because  and division by zero is undefined. On the Cartesian coordinate system, the function  would have a vertical asymptote at . In limit notation, this would be written as:

The limit of  as  approaches  is , which is denoted by  

 Right and left limits 
Consider the function , we can get as close to  in the -values as we want, so long as we do not make  equal to . For instance, we could make x=.00000001 or -.00000001, but never 0. Therefore, we can get  as close as we want to  or  depending on if we approach 0 from the right side or the left side. The left limit is the limit the function tends to if we only approach the target x-value from the left, for instance in the case of  when getting close to the 0 x-value from the left side, by using x-values that are smaller than 0, the limit would approach . In the same way, the right limit is the limit the function tends to if we only approach the target x-value from the right, for instance in the case of  when getting close to the 0 x-value from the right side, by using x-values that are larger than 0, the limit would approach .

 Related pages 

 Derivative (mathematics), a quantity defined as a limit of slopes
 Infinity
 Limit of a sequence

 References 

Category:Calculus
Category:Mathematical analysis