aarjaneiro/planetmath huggingface dataset:
limit examples

limit examples

Example 1.

Determine the limit

lim

x

→

0

⁡

tan

⁡

x

x

.  — Using the definition of

tan

and the limit

rule of product

(

http://planetmath.org/LimitRulesOfFunctions

) we can write

lim

x

→

0

⁡

tan

⁡

x

x

=

lim

x

→

0

⁡

(

sin

⁡

x

x

⋅

1

cos

⁡

x

)

=

lim

x

→

0

⁡

sin

⁡

x

x

⋅

lim

x

→

0

⁡

1

cos

⁡

x

.

The limit in the former factor (

http://planetmath.org/

Product

) is 1 by the parent entry (

http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0

). Also the latter limit is 1, since

cos

⁡

x

and thus the quotient

1

cos

⁡

x

is

continuous

in the point

x

=

0

(see

continuity of sine and cosine

).  Accordingly the desired limit is

1

.

Example 2.

Determine the limit

lim

x

→

0

⁡

sin

⁡

a

⁢

x

sin

⁡

b

⁢

x

.  — As above, we can write

lim

x

→

0

⁡

sin

⁡

a

⁢

x

sin

⁡

b

⁢

x

=

lim

x

→

0

⁡

(

sin

⁡

a

⁢

x

a

⁢

x

⋅

b

⁢

x

sin

⁡

b

⁢

x

⋅

a

b

)

=

lim

x

→

0

⁡

sin

⁡

a

⁢

x

a

⁢

x

⋅

lim

x

→

0

⁡

b

⁢

x

sin

⁡

b

⁢

x

⋅

lim

x

→

0

⁡

a

b

=

1

⋅

1

⋅

a

b

=

a

b

.

Example 3.

The

perimeter

of a

regular

n

-gon,

circumscribed

to a circle with radius 1, is

2

⁢

n

⁢

tan

⁡

π

n

.  Determine the limit of this perimeter as

n

tends to

infinity

.  — Utilising the example 1 we can calculate

lim

n

→

∞

⁡

2

⁢

n

⁢

tan

⁡

π

n

=

lim

n

→

∞

⁡

2

⁢

π

⁢

tan

⁡

π

n

π

n

=

2

⁢

π

⋅

1

=

2

⁢

π

,

which is the

circumference

of the circle.

Example 4.

Determine the limit

lim

x

→

0

⁡

arcsin

⁡

x

x

.  — If we denote

arcsin

⁡

x

:=

y

,

the

monotonicity

of the arcus sine (

http://planetmath.org/CyclometricFunctions

)

function

on

[

-

1

,

1

]

implies that “

x

→

0

” is

equivalent

(

http://planetmath.org/Equivalent3

) to “

y

→

0

”.  Then we can calculate:

lim

x

→

0

⁡

arcsin

⁡

x

x

=

lim

y

→

0

⁡

y

sin

⁡

y

=

lim

y

→

0

⁡

1

sin

⁡

y

y

=

1

1

=

1

.

Example 5.

One may use the definition of

derivative

in

lim

x

→

0

⁡

arctan

⁡

x

x

=

lim

x

→

0

⁡

arctan

⁡

x

-

arctan

⁡

0

x

-

0

=

[

d

d

⁢

x

⁢

arctan

⁡

x

]

x

=

0

=

1

1

+

0

2

=

1

.

Title

limit examples

Canonical name

LimitExamples

Date of creation

2013-03-22 17:40:16

Last modified on

2013-03-22 17:40:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 26A06

Classification

msc 26A03

Synonym

utilizing limit of

sin

⁡

x

x

in 0

Related topic

LimitRulesOfFunctions

Related topic

DerivativeOfInverseFunction

Related topic

ListOfCommonLimits

Generated on Fri Feb 9 19:44:03 2018 by

LaTeXML
limit of {1-c⁢o⁢sx}/x as x approaches 0

limit of

1

-

c

⁢

o

⁢

s

x

x

as

x

approaches 0

Corollary 1

.

lim

x

→

0

⁡

1

-

cos

⁡

x

x

=

0

Proof.

lim

x

→

0

⁡

1

-

cos

⁡

x

x

=

lim

x

→

0

⁡

(

1

-

cos

⁡

x

)

⁢

(

1

+

cos

⁡

x

)

x

⁢

(

1

+

cos

⁡

x

)

=

lim

x

→

0

⁡

1

-

cos

⁡

x

+

cos

⁡

x

-

cos

2

⁡

x

x

⁢

(

1

+

cos

⁡

x

)

=

lim

x

→

0

⁡

1

-

cos

2

⁡

x

x

⁢

(

1

+

cos

⁡

x

)

=

lim

x

→

0

⁡

sin

2

⁡

x

x

⁢

(

1

+

cos

⁡

x

)

=

lim

x

→

0

⁡

sin

⁡

x

x

⋅

sin

⁡

x

1

+

cos

⁡

x

=

(

lim

x

→

0

⁡

sin

⁡

x

x

)

⁢

(

lim

x

→

0

⁡

sin

⁡

x

1

+

cos

⁡

x

)

by this entry (

http://planetmath.org/LimitRulesOfFunctions

)

=

1

⋅

lim

x

→

0

⁡

sin

⁡

x

1

+

cos

⁡

x

by this entry (

http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0

)

=

sin

⁡

0

1

+

cos

⁡

0

by this entry (

http://planetmath.org/LimitRulesOfFunctions

) and the fact that

sin

and

cos

are

continuous

=

0

1

+

1

=

0

∎

This corollary has an obvious corollary to it:

Corollary 2

.

lim

x

→

0

⁡

cos

⁡

x

-

1

x

=

0

Title

limit of

1

-

c

⁢

o

⁢

s

x

x

as

x

approaches 0

Canonical name

LimitOfdisplaystylefrac1cosXxAsXApproaches0

Date of creation

2013-03-22 16:58:49

Last modified on

2013-03-22 16:58:49

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Corollary

Classification

msc 26A06

Classification

msc 26A03

Related topic

DerivativesOfSinXAndCosX

Generated on Fri Feb 9 19:44:06 2018 by

LaTeXML
limit of {s⁢i⁢nx}/x as x approaches 0

limit of

s

⁢

i

⁢

n

x

x

as

x

approaches 0

Theorem 1

.

lim

x

→

0

⁡

sin

⁡

x

x

=

1

x

<

tan

⁡

x

for

0

<

x

<

π

2

.

Proof.

First, let

0

<

x

<

π

2

. Then

0

<

cos

⁡

x

<

1

. Note also that

x

<

tan

⁡

x

.

(1)

Multiplying both of this

inequality

by

cos

⁡

x

yields

x

⁢

cos

⁡

x

<

sin

⁡

x

.

(2)

By this theorem (

http://planetmath.org/ComparisonOfSinThetaAndThetaNearTheta0

),

sin

⁡

x

<

x

.

(3)

Combining inequalities (

2

) and (

3

) gives

x

⁢

cos

⁡

x

<

sin

⁡

x

<

x

.

(4)

Dividing by

x

yields

cos

⁡

x

<

sin

⁡

x

x

<

1

.

(5)

Now let

-

π

2

<

x

<

0

. Then

0

<

-

x

<

π

2

. Plugging

-

x

into inequality (

5

) gives

cos

⁡

(

-

x

)

<

sin

⁡

(

-

x

)

-

x

<

1

.

(6)

Since

cos

is an

even function

and

sin

is an

odd function

, we have

cos

⁡

x

<

-

sin

⁡

x

-

x

<

1

.

(7)

Therefore, inequality (

5

) holds for all real

x

with

0

<

|

x

|

<

π

2

.

Since

cos

is

continuous

,

lim

x

→

0

⁡

cos

⁡

x

=

cos

⁡

0

=

1

. Thus,

1

=

lim

x

→

0

⁡

cos

⁡

x

≤

lim

x

→

0

⁡

sin

⁡

x

x

≤

lim

x

→

0

⁡

1

=

1

.

(8)

By the

squeeze theorem

, it follows that

lim

x

→

0

⁡

sin

⁡

x

x

=

1

.
∎

Note that the above limit is also valid if

x

is considered as a complex variable.

Title

limit of

s

⁢

i

⁢

n

x

x

as

x

approaches 0

Canonical name

LimitOfdisplaystylefracsinXxAsXApproaches0

Date of creation

2013-03-22 16:58:45

Last modified on

2013-03-22 16:58:45

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 26A06

Classification

msc 26A03

Related topic

ComparisonOfSinThetaAndThetaNearTheta0

Related topic

SincFunction

Related topic

DerivativesOfSinXAndCosX

Related topic

DerivativesOfSineAndCosine

Generated on Fri Feb 9 19:44:09 2018 by

LaTeXML
limit rules of functions

limit rules of functions

Theorem 1

.

Let

f

and

g

be two real (

http://planetmath.org/RealFunction

) or

complex functions

.  Suppose that there exist the limits

lim

x

→

x

0

⁡

f

⁢

(

x

)

and

lim

x

→

x

0

⁡

g

⁢

(

x

)

.  Then there exist the limits

lim

x

→

x

0

⁡

[

f

⁢

(

x

)

±

g

⁢

(

x

)

]

,

lim

x

→

x

0

⁡

f

⁢

(

x

)

⁢

g

⁢

(

x

)

and, if

lim

x

→

x

0

⁡

g

⁢

(

x

)

≠

0

,  also

lim

x

→

x

0

⁡

f

⁢

(

x

)

/

g

⁢

(

x

)

, and

1.

lim

x

→

x

0

⁡

[

f

⁢

(

x

)

±

g

⁢

(

x

)

]

=

lim

x

→

x

0

⁡

f

⁢

(

x

)

±

lim

x

→

x

0

⁡

g

⁢

(

x

)

,

2.

lim

x

→

x

0

⁡

f

⁢

(

x

)

⁢

g

⁢

(

x

)

=

lim

x

→

x

0

⁡

f

⁢

(

x

)

⋅

lim

x

→

x

0

⁡

g

⁢

(

x

)

,

3.

lim

x

→

x

0

⁡

f

⁢

(

x

)

g

⁢

(

x

)

=

lim

x

→

x

0

⁡

f

⁢

(

x

)

lim

x

→

x

0

⁡

g

⁢

(

x

)

,

4.

lim

x

→

x

0

⁡

c

=

c

where

⁢

c

⁢

is

⁢

a

⁢

constant

.

These rules are used in limit calculations and in proving the corresponding differentiation rules (

sum rule

,

product rule

etc.).

In 1, the domains of

f

and

g

could be any

topological space

(not necessarily

ℝ

or

ℂ

).

There are

limit rules of sequences

(

http://planetmath.org/Sequence

).

As well, one often needs the

Theorem 2

.

If there exists the limit

lim

x

→

x

0

⁡

f

⁢

(

x

)

=

a

and if

g

is

continuous at

the point

x

=

a

, then there exists the limit

lim

x

→

x

0

⁡

g

⁢

(

f

⁢

(

x

)

)

, and

lim

x

→

x

0

⁡

g

⁢

(

f

⁢

(

x

)

)

=

g

⁢

(

lim

x

→

x

0

⁡

f

⁢

(

x

)

)

.

Title

limit rules of functions

Canonical name

LimitRulesOfFunctions

Date of creation

2013-03-22 14:51:21

Last modified on

2013-03-22 14:51:21

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

20

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A06

Classification

msc 30A99

Synonym

limit rules of sequences

Related topic

GrowthOfExponentialFunction

Related topic

ImproperLimits

Related topic

DerivativesOfSineAndCosine

Related topic

ListOfCommonLimits

Related topic

LimitExamples

Related topic

ProductAndQuotientOfFunctionsSum

Related topic

DerivationOfPlasticNumber

Generated on Fri Feb 9 19:44:11 2018 by

LaTeXML
logarithmic proof of product rule

logarithmic proof of product rule

Following is a proof of the

product rule

using the

natural logarithm

, the

chain rule

, and

implicit differentiation

. Note that circular reasoning does not occur, as each of the concepts used can be proven independently of the product rule.

Proof.

Let

f

and

g

be

differentiable functions

and

y

=

f

⁢

(

x

)

⁢

g

⁢

(

x

)

. Then

ln

⁡

y

=

ln

⁡

(

f

⁢

(

x

)

⁢

g

⁢

(

x

)

)

=

ln

⁡

f

⁢

(

x

)

+

ln

⁡

g

⁢

(

x

)

. Thus,

1

y

⋅

d

⁢

y

d

⁢

x

=

f

′

⁢

(

x

)

f

⁢

(

x

)

+

g

′

⁢

(

x

)

g

⁢

(

x

)

. Therefore,

d

⁢

y

d

⁢

x

=

y

⁢

(

f

′

⁢

(

x

)

f

⁢

(

x

)

+

g

′

⁢

(

x

)

g

⁢

(

x

)

)

=

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

(

f

′

⁢

(

x

)

f

⁢

(

x

)

+

g

′

⁢

(

x

)

g

⁢

(

x

)

)

=

f

′

⁢

(

x

)

⁢

g

⁢

(

x

)

+

g

′

⁢

(

x

)

⁢

f

⁢

(

x

)

.

∎

Once students are familiar with the natural logarithm, the chain rule, and implicit differentiation, they typically have no problem following this proof of the product rule. Actually, with some prompting, they can produce a proof of the product rule to this one. This exercise is a great way for students to review many concepts from

calculus

.

Title

logarithmic proof of product rule

Canonical name

LogarithmicProofOfProductRule

Date of creation

2013-03-22 16:18:48

Last modified on

2013-03-22 16:18:48

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 26A06

Classification

msc 97D40

Related topic

LogarithmicProofOfQuotientRule

Generated on Fri Feb 9 19:44:17 2018 by

LaTeXML
logarithmic proof of quotient rule

logarithmic proof of quotient rule

Following is a proof of the

quotient rule

using the

natural logarithm

, the

chain rule

, and

implicit differentiation

. Note that circular reasoning does not occur, as each of the concepts used can be proven independently of the quotient rule.

Proof.

Let

f

and

g

be

differentiable functions

and

y

=

f

⁢

(

x

)

g

⁢

(

x

)

. Then

ln

⁡

y

=

ln

⁡

f

⁢

(

x

)

-

ln

⁡

g

⁢

(

x

)

. Thus,

1

y

⋅

d

⁢

y

d

⁢

x

=

f

′

⁢

(

x

)

f

⁢

(

x

)

-

g

′

⁢

(

x

)

g

⁢

(

x

)

. Therefore,

d

⁢

y

d

⁢

x

=

y

⁢

(

f

′

⁢

(

x

)

f

⁢

(

x

)

-

g

′

⁢

(

x

)

g

⁢

(

x

)

)

=

f

⁢

(

x

)

g

⁢

(

x

)

⁢

(

f

′

⁢

(

x

)

f

⁢

(

x

)

-

g

′

⁢

(

x

)

g

⁢

(

x

)

)

=

f

′

⁢

(

x

)

g

⁢

(

x

)

-

f

⁢

(

x

)

⁢

g

′

⁢

(

x

)

(

g

⁢

(

x

)

)

2

=

g

⁢

(

x

)

⁢

f

′

⁢

(

x

)

-

f

⁢

(

x

)

⁢

g

′

⁢

(

x

)

(

g

⁢

(

x

)

)

2

.

∎

Once students are familiar with the natural logarithm, the chain rule, and implicit differentiation, they typically have no problem following this proof of the quotient rule. Actually, with some prompting, they can produce a proof of the quotient rule to this one. This exercise is a great way for students to review many concepts from

calculus

.

Title

logarithmic proof of quotient rule

Canonical name

LogarithmicProofOfQuotientRule

Date of creation

2013-03-22 16:18:51

Last modified on

2013-03-22 16:18:51

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 26A06

Classification

msc 97D40

Related topic

LogarithmicProofOfProductRule

Generated on Fri Feb 9 19:44:20 2018 by

LaTeXML
mean square deviation

mean square deviation

If

f

is a

Riemann integrable

real function

on the

interval

[

a

,

b

]

which is wished to be approximated by another

function

φ

with the same

property

, then the mean (

http://planetmath.org/MeanValueTheorem

)

m

=

1

b

-

a

⁢

∫

a

b

[

f

⁢

(

x

)

-

φ

⁢

(

x

)

]

2

⁢

𝑑

x

is called the

mean square deviation

of

φ

from

f

.

For example, if

sin

⁡

x

is approximated by

x

on

[

0

,

π

2

]

, the mean square deviation is

2

π

⁢

∫

0

π

2

(

sin

⁡

x

-

x

)

2

⁢

𝑑

x

≈

0.04923

.

Title

mean square deviation

Canonical

name

MeanSquareDeviation

Date of creation

2013-03-22 18:21:57

Last modified on

2013-03-22 18:21:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A06

Classification

msc 41A99

Classification

msc 26A42

Synonym

mean squared error

Related topic

Variance

Related topic

RmsError

Related topic

AverageValueOfFunction

Generated on Fri Feb 9 19:44:30 2018 by

LaTeXML
mean-value theorem

mean-value theorem

Let

f

:

ℝ

→

ℝ

be a function which is

continuous

on the

interval

[

a

,

b

]

and

differentiable

on

(

a

,

b

)

. Then there exists a number

c

:

a

<

c

<

b

such that

f

′

⁢

(

c

)

=

f

⁢

(

b

)

-

f

⁢

(

a

)

b

-

a

.

(1)

The geometrical meaning of this theorem is illustrated in the picture:

The dashed line connects the points

(

a

,

f

⁢

(

a

)

)

and

(

b

,

f

⁢

(

b

)

)

. There is

c

between

a

and

b

at which the

tangent

to

f

has the same slope as the dashed line.

The

mean-value theorem

is often used in the

integral

context: There is a

c

∈

[

a

,

b

]

such that

(

b

-

a

)

⁢

f

⁢

(

c

)

=

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

.

(2)

Title

mean-value theorem

Canonical name

MeanvalueTheorem

Date of creation

2013-03-22 12:20:39

Last modified on

2013-03-22 12:20:39

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

9

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 26A06

Related topic

RollesTheorem

Related topic

IntermediateValueTheorem

Related topic

ExtendedMeanValueTheorem

Related topic

ProofOfExtendedMeanValueTheorem

Related topic

DerivationOfWaveEquation

Generated on Fri Feb 9 19:44:32 2018 by

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mean-value theorem for several variables

mean-value theorem for several variables

The

mean-value theorem

for a function of one real variable may be generalised for functions of arbitrarily many real variables; for the sake of concreteness, we here formulate it for the case of three variables:

Theorem.

If a function

f

⁢

(

x

,

y

,

z

)

is

continuously differentiable

in an

open set

of

ℝ

3

containing the points

(

x

1

,

y

1

,

z

1

)

and

(

x

2

,

y

2

,

z

2

)

and the

line segment

connecting them, then an equation

f

⁢

(

x

2

,

y

2

,

z

2

)

-

f

⁢

(

x

1

,

y

1

,

z

1

)

=

f

x

′

⁢

(

a

,

b

,

c

)

⁢

(

x

2

-

x

1

)

+

f

y

′

⁢

(

a

,

b

,

c

)

⁢

(

y

2

-

y

1

)

+

f

z

′

⁢

(

a

,

b

,

c

)

⁢

(

z

2

-

z

1

)

,

where

(

a

,

b

,

c

)

an

interior point

of the line segment, is .

Title

mean-value theorem for several variables

Canonical name

MeanvalueTheoremForSeveralVariables

Date of creation

2013-03-22 19:11:36

Last modified on

2013-03-22 19:11:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A06

Classification

msc 26B05

Generated on Fri Feb 9 19:44:34 2018 by

LaTeXML
monotonicity criterion

monotonicity criterion

Suppose that

f

:

[

a

,

b

]

→

ℝ

is a

function

which is

continuous

on

[

a

,

b

]

and

differentiable

on

(

a

,

b

)

.

Then the following relations hold.

1.

f

′

⁢

(

x

)

≥

0

for all

x

∈

(

a

,

b

)

⇔

f

is an

increasing function

on

[

a

,

b

]

;

2.

f

′

⁢

(

x

)

≤

0

for all

x

∈

(

a

,

b

)

⇔

f

is a

decreasing function

on

[

a

,

b

]

;

3.

f

′

⁢

(

x

)

>

0

for all

x

∈

(

a

,

b

)

⇒

f

is a

strictly increasing function

on

[

a

,

b

]

;

4.

f

′

⁢

(

x

)

<

0

for all

x

∈

(

a

,

b

)

⇒

f

is a

strictly decreasing function

on

[

a

,

b

]

.

Notice that the third and fourth statement cannot be inverted. As an example consider the function

f

:

[

-

1

,

1

]

→

ℝ

,

f

⁢

(

x

)

=

x

3

. This is a strictly increasing function, but

f

′

⁢

(

0

)

=

0

.

Title

monotonicity criterion

Canonical name

MonotonicityCriterion

Date of creation

2013-03-22 13:45:12

Last modified on

2013-03-22 13:45:12

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

7

Author

paolini (1187)

Entry type

Theorem

Classification

msc 26A06

Generated on Fri Feb 9 19:44:36 2018 by

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nabla

nabla

Let

f

:

ℝ

n

→

ℝ

be a

C

1

⁢

(

ℝ

n

)

function

, that is, a partially

differentiable
function

in all its coordinates. The symbol

∇

, named

nabla

, represents the

gradient

operator

, whose action on

f

⁢

(

x

1

,

x

2

,

…

,

x

n

)

is given by

∇

⁡

f

=

(

f

x

1

,

f

x

2

,

…

,

f

x

n

)

=

(

∂

⁡

f

∂

⁡

x

1

,

∂

⁡

f

∂

⁡

x

2

,

…

,

∂

⁡

f

∂

⁡

x

n

)

Properties

1.

If

f

,

g

are functions, then

∇

⁡

(

f

⁢

g

)

=

(

∇

⁡

f

)

⁢

g

+

f

⁢

∇

⁡

g

.

2.

For any scalars

α

and

β

and functions

f

and

g

,

∇

⁡

(

α

⁢

f

+

β

⁢

g

)

=

α

⁢

∇

⁡

f

+

β

⁢

∇

⁡

g

.

The

∇

symbolism

Using the

∇

formalism

,
the

divergence

operator can be expressed as

∇

⋅

, the curl operator as

∇

×

, and the
Laplacian operator as

∇

2

. To wit, for a given vector field

𝐀

=

A

x

⁢

𝐢

+

A

y

⁢

𝐣

+

A

z

⁢

𝐤

,

and a given function

f

we have

∇

⋅

𝐀

=

∂

⁡

A

x

∂

⁡

x

+

∂

⁡

A

y

∂

⁡

y

+

∂

⁡

A

z

∂

⁡

z

∇

×

𝐀

=

(

∂

⁡

A

z

∂

⁡

y

-

∂

⁡

A

y

∂

⁡

z

)

⁢

𝐢

+

(

∂

⁡

A

x

∂

⁡

z

-

∂

⁡

A

z

∂

⁡

x

)

⁢

𝐣

+

(

∂

⁡

A

y

∂

⁡

x

-

∂

⁡

A

x

∂

⁡

y

)

⁢

𝐤

∇

2

⁡

f

=

∂

2

⁡

f

∂

⁡

x

2

+

∂

2

⁡

f

∂

⁡

y

2

+

∂

2

⁡

f

∂

⁡

z

2

.

Title

nabla

Canonical name

Nabla

Date of creation

2013-03-22 14:00:20

Last modified on

2013-03-22 14:00:20

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

7

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 26A06

Related topic

gradient

Related topic

NablaActingOnProducts

Related topic

Gradient

Related topic

AlternateCharacterizationOfCurl

Defines

∇

Generated on Fri Feb 9 19:44:38 2018 by

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one-sided continuity

one-sided continuity

The

real function

f

is

continuous

from the left

in the point

x

=

x

0

iff

lim

x

→

x

0

-

⁡

f

⁢

(

x

)

=

f

⁢

(

x

0

)

.

The real function

f

is

continuous from the right

in the point

x

=

x

0

iff

lim

x

→

x

0

+

⁡

f

⁢

(

x

)

=

f

⁢

(

x

0

)

.

The real function

f

is

continuous on the

closed interval

[

a

,

b

]

iff it is continuous at all points of the

open interval

(

a

,

b

)

,  from the right continuous at

a

and from the left continuous at

b

.

Examples.

The

ceiling function

⌈

x

⌉

is from the left continuous at each

integer

, the

mantissa function

x

-

⌊

x

⌋

is from the right continuous at each integer.

Title

one-sided continuity

Canonical name

OnesidedContinuity

Date of creation

2013-03-22 17:57:50

Last modified on

2013-03-22 17:57:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A06

Related topic

OneSidedLimit

Related topic

OneSidedDerivatives

Related topic

OneSidedContinuityBySeries

Defines

continuous from the left

Defines

continuous from the right

Defines

from the left continuous

Defines

from the right continuous

Defines

continuous on closed interval

Generated on Fri Feb 9 19:44:56 2018 by

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one-sided limit

one-sided limit

Definition

Let

f

be a real-valued function defined on

S

⊆

ℝ

. The

left-hand

one-sided
limit

at

a

∈

ℝ

is defined to be the

real number

L

-

such that for every

ϵ

>

0

there
exists a

δ

>

0

such that

|

f

⁢

(

x

)

-

L

-

|

<

ϵ

whenever

0

<

a

-

x

<

δ

.

Analogously, the

right-hand one-sided limit

at

a

∈

ℝ

is the
real number

L

+

such that
for every

ϵ

>

0

there exists a

δ

>

0

such that

|

f

⁢

(

x

)

-

L

+

|

<

ϵ

whenever

0

<

x

-

a

<

δ

.

Common notations for the one-sided limits are

L

+

=

f

⁢

(

x

+

)

=

lim

x

→

a

+

⁡

f

⁢

(

x

)

=

lim

x

↘

a

⁡

f

⁢

(

x

)

,

L

-

=

f

⁢

(

x

-

)

=

lim

x

→

a

-

⁡

f

⁢

(

x

)

=

lim

x

↗

a

⁡

f

⁢

(

x

)

.

Sometimes, left-handed limits are referred to as limits

from below

while
right-handed limits are

from above

.

Theorem

The ordinary limit of a function exists at a point if and only
if both one-sided limits exist at this point and are equal (to the ordinary limit).

Example

The Heaviside unit step function, sometimes colloquially referred to as the diving board function,
defined by

H

⁢

(

x

)

=

{

0

if

⁢

x

<

0

1

if

⁢

x

≥

0

has the simplest kind of

discontinuity

at

x

=

0

, a

jump discontinuity

. Its ordinary limit does not exist at this point, but the one-sided limits do exist, and are

lim

x

→

0

-

⁡

H

⁢

(

x

)

=

0

⁢

and

⁢

lim

x

→

0

+

⁡

H

⁢

(

x

)

=

1

.

Title

one-sided limit

Canonical

name

OnesidedLimit

Date of creation

2013-03-22 12:40:28

Last modified on

2013-03-22 12:40:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 26A06

Synonym

limit from below

Synonym

limit from above

Synonym

left-sided limit

Synonym

left-handed limit

Synonym

right-sided limit

Synonym

right-handed limit

Related topic

Limit

Related topic

OneSidedDerivatives

Related topic

IntegratingTanXOver0fracpi2

Related topic

OneSidedContinuity

Defines

Heaviside unit step function

Generated on Fri Feb 9 19:44:59 2018 by

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oscillation of a function

oscillation of a function

Definition 1

.

Let

f

:

X

⊂

R

→

R

. The

oscillation

of the

function

f

on the set

X

is said to be

ω

⁢

(

f

,

X

)

=

sup

a

,

b

∈

X

⁡

|

f

⁢

(

b

)

-

f

⁢

(

a

)

|

,

where

a

,

b

are arbitrary points in

X

.

0.1

Examples

•

ω

⁢

(

x

2

,

[

-

1

,

2

]

)

=

4

•

ω

⁢

(

x

,

[

-

1

,

2

]

)

=

3

•

ω

⁢

(

x

,

(

-

1

,

2

)

)

=

3

•

ω

⁢

(

sgn

⁡

x

⁢

[

-

1

,

2

]

)

=

2

•

ω

⁢

(

sgn

⁡

x

⁢

[

0

,

2

]

)

=

1

•

ω

⁢

(

sgn

⁡

x

⁢

(

0

,

2

]

)

=

0

Cauchy’s criterion can be expressed in terms of this concept.

[

1

]

References

1

V., Zorich,

Mathematical Analysis

I

, pp. 131, First Ed., Springer-Verlag, 2004.

Title

oscillation of a function

Canonical name

OscillationOfAFunction

Date of creation

2013-03-22 17:45:50

Last modified on

2013-03-22 17:45:50

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

5

Author

perucho (2192)

Entry type

Definition

Classification

msc 26A06

Related topic

TotalVariation

Generated on Fri Feb 9 19:45:01 2018 by

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