20 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| List of limits | 3/3 | https://en.wikipedia.org/wiki/List_of_limits | reference | science, encyclopedia | 2026-05-05T08:15:28.347754+00:00 | kb-cron |
lim
x
→
1
ln
(
x
)
x
−
1
=
1
{\displaystyle \lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1}
lim
x
→
0
ln
(
x
+
1
)
x
=
1
{\displaystyle \lim _{x\to 0}{\frac {\ln(x+1)}{x}}=1}
lim
x
→
0
−
ln
(
1
+
a
(
e
−
x
−
1
)
)
x
=
a
{\displaystyle \lim _{x\to 0}{\frac {-\ln \left(1+a\left({e^{-x}-1}\right)\right)}{x}}=a}
. This limit follows from L'Hôpital's rule.
lim
x
→
0
x
ln
x
=
0
{\displaystyle \lim _{x\to 0}x\ln x=0}
, hence
lim
x
→
0
x
x
=
1
{\displaystyle \lim _{x\to 0}x^{x}=1}
lim
x
→
∞
ln
x
x
=
0
{\displaystyle \lim _{x\to \infty }{\frac {\ln x}{x}}=0}
=== Logarithms to arbitrary bases === For b > 1,
lim
x
→
0
+
log
b
x
=
−
∞
{\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-\infty }
lim
x
→
∞
log
b
x
=
∞
{\displaystyle \lim _{x\to \infty }\log _{b}x=\infty }
For b < 1,
lim
x
→
0
+
log
b
x
=
∞
{\displaystyle \lim _{x\to 0^{+}}\log _{b}x=\infty }
lim
x
→
∞
log
b
x
=
−
∞
{\displaystyle \lim _{x\to \infty }\log _{b}x=-\infty }
Both cases can be generalized to:
lim
x
→
0
+
log
b
x
=
−
F
(
b
)
∞
{\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-F(b)\infty }
lim
x
→
∞
log
b
x
=
F
(
b
)
∞
{\displaystyle \lim _{x\to \infty }\log _{b}x=F(b)\infty }
where
F
(
x
)
=
2
H
(
x
−
1
)
−
1
{\displaystyle F(x)=2H(x-1)-1}
and
H
(
x
)
{\displaystyle H(x)}
is the Heaviside step function
== 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}
. 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}
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 x0 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. x0 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.
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.
== Notable special limits ==
lim
n
→
∞
n
n
!
n
=
e
{\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}
lim
n
→
∞
(
n
!
)
1
/
n
=
∞
{\displaystyle \lim _{n\to \infty }\left(n!\right)^{1/n}=\infty }
. This can be proven by considering the inequality
e
x
≥
x
n
n
!
{\displaystyle e^{x}\geq {\frac {x^{n}}{n!}}}
at
x
=
n
{\displaystyle x=n}
.
lim
n
→
∞
2
n
2
−
2
+
2
+
⋯
+
2
⏟
n
=
π
{\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\dots +{\sqrt {2}}}}}}}} _{n}=\pi }
. This can be derived from Viète's formula for π.
== Limiting behavior ==
=== Asymptotic equivalences === Asymptotic equivalences,
f
(
x
)
∼
g
(
x
)
{\displaystyle f(x)\sim g(x)}
, are true if
lim
x
→
∞
f
(
x
)
g
(
x
)
=
1
{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1}
. Therefore, they can also be reframed as limits. Some notable asymptotic equivalences include
lim
x
→
∞
x
/
ln
x
π
(
x
)
=
1
{\displaystyle \lim _{x\to \infty }{\frac {x/\ln x}{\pi (x)}}=1}
, due to the prime number theorem,
π
(
x
)
∼
x
ln
x
{\displaystyle \pi (x)\sim {\frac {x}{\ln x}}}
, where π(x) is the prime counting function.
lim
n
→
∞
2
π
n
(
n
e
)
n
n
!
=
1
{\displaystyle \lim _{n\to \infty }{\frac {{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}{n!}}=1}
, due to Stirling's approximation,
n
!
∼
2
π
n
(
n
e
)
n
{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}
.
=== Big O notation === The behaviour of functions described by Big O notation can also be described by limits. For example
f
(
x
)
∈
O
(
g
(
x
)
)
{\displaystyle f(x)\in {\mathcal {O}}(g(x))}
if
lim sup
x
→
∞
|
f
(
x
)
|
g
(
x
)
<
∞
{\displaystyle \limsup _{x\to \infty }{\frac {|f(x)|}{g(x)}}<\infty }
== References ==