16 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| List of limits | 1/3 | https://en.wikipedia.org/wiki/List_of_limits | reference | science, encyclopedia | 2026-05-05T08:15:28.347754+00:00 | kb-cron |
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}
lim
x
→
c
1
f
(
x
)
=
1
L
{\displaystyle \lim _{x\to c}{\frac {1}{f(x)}}={\frac {1}{L}}}
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
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. 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)}
=== 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}}
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}}
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}
=== 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)}}}
=== 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
lim
x
→
c
f
(
x
)
≤
lim
x
→
c
g
(
x
)
{\displaystyle \lim _{x\to c}f(x)\leq \lim _{x\to c}g(x)}