This is a list of limits for common functions such as elementary functions . In this article, the terms a , b and c are constants with respect to x .
Limits for general functions
Definitions of limits and related concepts
lim
x
→
c
f
(
x
)
=
L
{\displaystyle \lim _{x\to c}f(x)=L}
if and only if
∀
ε
>
0
∃
δ
>
0
:
0
<
|
x
−
c
|
<
δ
⟹
|
f
(
x
)
−
L
|
<
ε
{\displaystyle \forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon }
. This is the (ε, δ)-definition of limit .
The limit superior and limit inferior of a sequence are defined as
lim sup
n
→
∞
x
n
=
lim
n
→
∞
(
sup
m
≥
n
x
m
)
{\displaystyle \limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)}
and
lim inf
n
→
∞
x
n
=
lim
n
→
∞
(
inf
m
≥
n
x
m
)
{\displaystyle \liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)}
.
A function,
f
(
x
)
{\displaystyle f(x)}
, is said to be continuous at a point, c , if
lim
x
→
c
f
(
x
)
=
f
(
c
)
.
{\displaystyle \lim _{x\to c}f(x)=f(c).}
Operations on a single known limit
If
lim
x
→
c
f
(
x
)
=
L
{\displaystyle \lim _{x\to c}f(x)=L}
then:
lim
x
→
c
[
f
(
x
)
±
a
]
=
L
±
a
{\displaystyle \lim _{x\to c}\,[f(x)\pm a]=L\pm a}
lim
x
→
c
a
f
(
x
)
=
a
L
{\displaystyle \lim _{x\to c}\,af(x)=aL}
[1] [2] [3]
lim
x
→
c
1
f
(
x
)
=
1
L
{\displaystyle \lim _{x\to c}{\frac {1}{f(x)))={\frac {1}{L))}
[4] if L is not equal to 0.
lim
x
→
c
f
(
x
)
n
=
L
n
{\displaystyle \lim _{x\to c}\,f(x)^{n}=L^{n))
if n is a positive integer[1] [2] [3]
lim
x
→
c
f
(
x
)
1
n
=
L
1
n
{\displaystyle \lim _{x\to c}\,f(x)^{1 \over n}=L^{1 \over n))
if n is a positive integer, and if n is even, then L > 0.[1] [3] In general, if g (x ) is continuous at L and
lim
x
→
c
f
(
x
)
=
L
{\displaystyle \lim _{x\to c}f(x)=L}
then
lim
x
→
c
g
(
f
(
x
)
)
=
g
(
L
)
{\displaystyle \lim _{x\to c}g\left(f(x)\right)=g(L)}
[1] [2]
Operations on two known limits
If
lim
x
→
c
f
(
x
)
=
L
1
{\displaystyle \lim _{x\to c}f(x)=L_{1))
and
lim
x
→
c
g
(
x
)
=
L
2
{\displaystyle \lim _{x\to c}g(x)=L_{2))
then:
lim
x
→
c
[
f
(
x
)
±
g
(
x
)
]
=
L
1
±
L
2
{\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2))
[1] [2] [3]
lim
x
→
c
[
f
(
x
)
g
(
x
)
]
=
L
1
⋅
L
2
{\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\cdot L_{2))
[1] [2] [3]
lim
x
→
c
f
(
x
)
g
(
x
)
=
L
1
L
2
if
L
2
≠
0
{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)))={\frac {L_{1)){L_{2))}\qquad {\text{ if ))L_{2}\neq 0}
[1] [2] [3]
Limits involving derivatives or infinitesimal changes
In these limits, the infinitesimal change
h
{\displaystyle h}
is often denoted
Δ
x
{\displaystyle \Delta x}
or
δ
x
{\displaystyle \delta x}
. If
f
(
x
)
{\displaystyle f(x)}
is differentiable at
x
{\displaystyle x}
,
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
=
f
′
(
x
)
{\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)}
. This is the definition of the derivative . All differentiation rules can also be reframed as rules involving limits. For example, if g (x ) is differentiable at x ,
lim
h
→
0
f
∘
g
(
x
+
h
)
−
f
∘
g
(
x
)
h
=
f
′
[
g
(
x
)
]
g
′
(
x
)
{\displaystyle \lim _{h\to 0}{f\circ g(x+h)-f\circ g(x) \over h}=f'[g(x)]g'(x)}
. This is the chain rule .
lim
h
→
0
f
(
x
+
h
)
g
(
x
+
h
)
−
f
(
x
)
g
(
x
)
h
=
f
′
(
x
)
g
(
x
)
+
f
(
x
)
g
′
(
x
)
{\displaystyle \lim _{h\to 0}{f(x+h)g(x+h)-f(x)g(x) \over h}=f'(x)g(x)+f(x)g'(x)}
. This is the product rule .
lim
h
→
0
(
f
(
x
+
h
)
f
(
x
)
)
1
/
h
=
exp
(
f
′
(
x
)
f
(
x
)
)
{\displaystyle \lim _{h\to 0}\left({\frac {f(x+h)}{f(x)))\right)^{1/h}=\exp \left({\frac {f'(x)}{f(x)))\right)}
lim
h
→
0
(
f
(
e
h
x
)
f
(
x
)
)
1
/
h
=
exp
(
x
f
′
(
x
)
f
(
x
)
)
{\displaystyle \lim _{h\to 0}{\left({f(e^{h}x) \over {f(x)))\right)^{1/h))=\exp \left({\frac {xf'(x)}{f(x)))\right)}
If
f
(
x
)
{\displaystyle f(x)}
and
g
(
x
)
{\displaystyle g(x)}
are differentiable on an open interval containing c , except possibly c itself, and
lim
x
→
c
f
(
x
)
=
lim
x
→
c
g
(
x
)
=
0
or
±
∞
{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or ))\pm \infty }
, L'Hôpital's rule can be used:
lim
x
→
c
f
(
x
)
g
(
x
)
=
lim
x
→
c
f
′
(
x
)
g
′
(
x
)
{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)))=\lim _{x\to c}{\frac {f'(x)}{g'(x)))}
[2]
Inequalities
If
f
(
x
)
≤
g
(
x
)
{\displaystyle f(x)\leq g(x)}
for all x in an interval that contains c , except possibly c itself, and the limit of
f
(
x
)
{\displaystyle f(x)}
and
g
(
x
)
{\displaystyle g(x)}
both exist at c , then[5]
lim
x
→
c
f
(
x
)
≤
lim
x
→
c
g
(
x
)
{\displaystyle \lim _{x\to c}f(x)\leq \lim _{x\to c}g(x)}
If
lim
x
→
c
f
(
x
)
=
lim
x
→
c
h
(
x
)
=
L
{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L}
and
f
(
x
)
≤
g
(
x
)
≤
h
(
x
)
{\displaystyle f(x)\leq g(x)\leq h(x)}
for all x in an open interval that contains c , except possibly c itself,
lim
x
→
c
g
(
x
)
=
L
.
{\displaystyle \lim _{x\to c}g(x)=L.}
This is known as the squeeze theorem .[1] [2] This applies even in the cases that f (x ) and g (x ) take on different values at c , or are discontinuous at c .
Trigonometric functions
If
x
{\displaystyle x}
is expressed in radians:
lim
x
→
a
sin
x
=
sin
a
{\displaystyle \lim _{x\to a}\sin x=\sin a}
lim
x
→
a
cos
x
=
cos
a
{\displaystyle \lim _{x\to a}\cos x=\cos a}
These limits both follow from the continuity of sin and cos.
lim
x
→
0
sin
x
x
=
1
{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x))=1}
.[7] [8] Or, in general,
lim
x
→
0
sin
a
x
a
x
=
1
{\displaystyle \lim _{x\to 0}{\frac {\sin ax}{ax))=1}
, for a not equal to 0.
lim
x
→
0
sin
a
x
x
=
a
{\displaystyle \lim _{x\to 0}{\frac {\sin ax}{x))=a}
lim
x
→
0
sin
a
x
b
x
=
a
b
{\displaystyle \lim _{x\to 0}{\frac {\sin ax}{bx))={\frac {a}{b))}
, for b not equal to 0.
lim
x
→
∞
x
sin
(
1
x
)
=
1
{\displaystyle \lim _{x\to \infty }x\sin \left({\frac {1}{x))\right)=1}
lim
x
→
0
1
−
cos
x
x
=
lim
x
→
0
cos
x
−
1
x
=
0
{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x))=\lim _{x\to 0}{\frac {\cos x-1}{x))=0}
[4] [8] [9]
lim
x
→
0
1
−
cos
x
x
2
=
1
2
{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x^{2))}={\frac {1}{2))}
lim
x
→
n
±
tan
(
π
x
+
π
2
)
=
∓
∞
{\displaystyle \lim _{x\to n^{\pm ))\tan \left(\pi x+{\frac {\pi }{2))\right)=\mp \infty }
, for integer n .
lim
x
→
0
tan
x
x
=
1
{\displaystyle \lim _{x\to 0}{\frac {\tan x}{x))=1}
. Or, in general,
lim
x
→
0
tan
a
x
a
x
=
1
{\displaystyle \lim _{x\to 0}{\frac {\tan ax}{ax))=1}
, for a not equal to 0.
lim
x
→
0
tan
a
x
b
x
=
a
b
{\displaystyle \lim _{x\to 0}{\frac {\tan ax}{bx))={\frac {a}{b))}
, for b not equal to 0.
lim
n
→
∞
sin
sin
⋯
sin
(
x
0
)
⏟
n
=
0
{\displaystyle \lim _{n\to \infty }\ \underbrace {\sin \sin \cdots \sin(x_{0})} _{n}=0}
, where x 0 is an arbitrary real number.
lim
n
→
∞
cos
cos
⋯
cos
(
x
0
)
⏟
n
=
d
{\displaystyle \lim _{n\to \infty }\ \underbrace {\cos \cos \cdots \cos(x_{0})} _{n}=d}
, where d is the Dottie number . x 0 can be any arbitrary real number.
Sums
In general, any infinite series is the limit of its partial sums . For example, an analytic function is the limit of its Taylor series , within its radius of convergence .
lim
n
→
∞
∑
k
=
1
n
1
k
=
∞
{\displaystyle \lim _{n\to \infty }\sum _{k=1}^{n}{\frac {1}{k))=\infty }
. This is known as the harmonic series .[6]
lim
n
→
∞
(
∑
k
=
1
n
1
k
−
log
n
)
=
γ
{\displaystyle \lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k))-\log n\right)=\gamma }
. This is the Euler Mascheroni constant .