list of common limits

list of common limits

Following is a

list of common limits

used in

elementary

calculus

:

•

For any

real numbers

a

and

c

,

l

⁢

i

⁢

m

x

→

a

⁢

c

=

c

.

•

For any real numbers

a

and

n

,

lim

x

→

a

⁡

x

n

=

a

n

(proven here (

http://

planetmath

.org/ContinuityOfNaturalPower

) for

n

a

positive integer

)

•

lim

x

→

0

⁡

sin

⁡

x

x

=

1

(proven here (

http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0

))

•

lim

x

→

0

⁡

1

-

cos

⁡

x

x

=

0

(proven here (

http://planetmath.org/LimitOfDisplaystyleFrac1CosXxAsXApproaches0

))

•

lim

x

→

0

⁡

arcsin

⁡

x

x

=

1

(proven here (

http://planetmath.org/LimitExamples

))

•

lim

x

→

0

⁡

e

x

-

1

x

=

1

(proven here (

http://planetmath.org/DerivativeOfExponentialFunction

))

•

For

a

>

0

,

lim

x

→

0

⁡

a

x

-

1

x

=

ln

⁡

a

(proven here (

http://planetmath.org/LimitOfDisplaystyleFracax1xAsXApproaches0

)).

•

For

b

>

1

and

a

any real number,

lim

x

→

∞

⁡

x

a

b

x

=

0

(proven here (

http://planetmath.org/GrowthOfExponentialFunction

)).

•

lim

x

→

0

+

⁡

x

x

=

1

(proven here (

http://planetmath.org/FunctionXx

))

•

lim

x

→

0

+

⁡

x

⁢

ln

⁡

x

=

0

(proven here (

http://planetmath.org/GrowthOfExponentialFunction

))

•

lim

x

→

∞

⁡

ln

⁡

x

x

=

0

(proven here (

http://planetmath.org/GrowthOfExponentialFunction

))

•

lim

x

→

∞

⁡

x

1

x

=

1

(proven here (

http://planetmath.org/GrowthOfExponentialFunction

))

•

lim

x

→

±

∞

⁡

(

1

+

1

x

)

x

=

e

•

lim

x

→

0

⁡

(

1

+

x

)

1

x

=

e

•

lim

x

→

0

⁡

(

1

+

sin

⁡

x

)

1

x

=

e

(power of

e

, l’Hôpital’s rule (

http://planetmath.org/LHpitalsRule

))

•

lim

x

→

∞

⁡

(

x

-

x

2

-

a

2

)

=

0

(proven here (

http://planetmath.org/Hyperbola

))

•

For

a

>

0

and

n

a positive integer,

lim

x

→

a

⁡

x

-

a

x

n

-

a

n

=

1

n

⁢

a

n

-

1

.

•

lim

x

→

0

⁡

tan

⁡

x

-

sin

⁡

x

x

3

=

1

2

(by l’Hôpital’s rule (

http://planetmath.org/LHpitalsRule

))

•

For

q

>

0

,

lim

x

→

∞

⁡

(

log

⁡

x

)

p

x

q

=

0

•

tan

⁡

(

x

+

π

2

)

=

lim

ξ

→

π

2

⁡

tan

⁡

x

+

tan

⁡

ξ

1

-

tan

⁡

x

⁢

tan

⁡

ξ

=

lim

ξ

→

π

2

⁡

sec

2

⁡

ξ

-

tan

⁡

x

⁢

sec

2

⁡

ξ

=

-

cot

⁡

x

(by  l’Hôpital’s rule (

http://planetmath.org/LHpitalsRule

))

That is,

tan

⁡

x

⁢

tan

⁡

(

x

+

π

2

)

=

-

1

, which indicates orthogonality of the slopes represented by those

functions

.

•

For a real or complex

constant

c

and a variable

z

,

lim

n

→

∞

⁡

n

n

+

1

z

n

+

1

⁢

(

c

+

n

z

)

-

(

n

+

1

)

=

e

-

c

⁢

z

.

•

For

x

real (or complex),

lim

n

→

∞

⁡

n

⁢

(

x

n

-

1

)

=

log

⁡

x

(proven here (

http://planetmath.org/HalleysFormula

) for real

x

).

PlanetMath,

References

1

Catherine Roberts & Ray McLenaghan, “

Continuous

Mathematics” in

Standard Mathematical Tables and Formulae

ed. Daniel Zwillinger. Boca Raton: CRC Press (1996): 333, 5.1

Differential Calculus

Title

list of common limits

Canonical name

ListOfCommonLimits

Date of creation

2014-02-23 10:09:07

Last modified on

2014-02-23 10:09:07

Owner

Wkbj79 (1863)

Last modified by

pahio (2872)

Numerical id

29

Author

Wkbj79 (2872)

Entry type

Feature

Classification

msc 26A06

Classification

msc 26A03

Classification

msc 26-00

Related topic

LimitRulesOfFunctions

Related topic

ImproperLimits

Related topic

LimitExamples

Related topic

HalleysFormula

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