!x!!
![{\displaystyle \lim _{x\to p}f(x)=L,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6df19e0c4d23fbfc27e6d71c95d2d5693810e6fd)
![{\displaystyle \lim _{x\to p}f(x)=L,\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c20b82c2747f10d5adfdd33f69e08cefe2887d)
![{\displaystyle \lim _{x\to p^{+}}f(x)=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe0b84549e3caf6ce648960ff5955955c59605c)
![{\displaystyle \lim _{x\to p^{-}}f(x)=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef9ef5a1e71bd645d6d1c7eb2231d54a5debc51)
![{\displaystyle \lim _{x\to p}f(x)=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29dc2f1b0911fbcda671e42b7a9c137c3577daf9)
![{\displaystyle \lim _{x\to p}f(x)=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29dc2f1b0911fbcda671e42b7a9c137c3577daf9)
![{\displaystyle \lim _{x\to p}f(x)=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29dc2f1b0911fbcda671e42b7a9c137c3577daf9)
![{\displaystyle \lim _{x\to \infty }f(x)=L,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8a64d74dc879eca107de742a57f3afb88489f9)
if and only if for all
there exists S > 0 such that
whenever x > S.
![{\displaystyle \lim _{x\to -\infty }f(x)=L,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98d62ae4649105191e1a58c97a744c48c3a9ea87)
if and only if for all
there exists S < 0 such that
whenever x < S.
![{\displaystyle \lim _{x\to -\infty }e^{x}=0.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d6d0227da0a38e16ff4e5f58e113a6f6f88190)
![{\displaystyle \lim _{x\to a}f(x)=\infty ,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7394ebf134950fcd1c2101a6f879350466c9989)
if and only if for all
there exists
such that
whenever
.
![{\displaystyle \lim _{x\to \infty }f(x)=\infty ,\lim _{x\to a^{+}}f(x)=-\infty .\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a08a3a85a8a2aad7fa9f01e006045be205629ef7)
![{\displaystyle \lim _{x\to 0^{+}}\ln x=-\infty .\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5ccfba4b85ab88eb7dfecd6764e034c31fc8db)
![{\displaystyle \lim _{x\to 0^{+}}{1 \over x}=\infty ,\lim _{x\to \infty }{1 \over x}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8467ab1ab6a752cf2e16c12ff0866837ae178e)
The complex plane with metric
is also a metric space. There are two different types of limits when the complex-valued functions are considered.
![{\displaystyle \lim _{x\to p}f(x)=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29dc2f1b0911fbcda671e42b7a9c137c3577daf9)
if and only if for all e > 0 there exists a d > 0 such that for all real numbers x with
, then
.
![{\displaystyle \lim _{(x,y)\to (p,q)}f(x,y)=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38d6be6ff6c67fd73621247629398c5050af8368)
![{\displaystyle {\begin{matrix}\lim \limits _{x\to p}&(f(x)+g(x))&=&\lim \limits _{x\to p}f(x)+\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)-g(x))&=&\lim \limits _{x\to p}f(x)-\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)\cdot g(x))&=&\lim \limits _{x\to p}f(x)\cdot \lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)/g(x))&=&{\lim \limits _{x\to p}f(x)/\lim \limits _{x\to p}g(x)}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2eddd546da2cf143602fd57cdfac4da149bdb2)
, and
,
is not true. However, this "chain rule" does hold if, in addition, either f(d) = e (i. e. f is continuous at d) or g does not take the value d near c (i. e. there exists a
such that if
then
).
![{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f2fb52f5211c7b7aa69d9e75195afaab5b9d5b1)
![{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e68c22c5c70aa65f792c06b2cf2fb6e1ecfe16b)
![{\displaystyle \sin x<x<\tan x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a95ca507973030e1df308fe9f32926f225a6ab28)
![{\displaystyle 1<{\frac {x}{\sin x}}<{\frac {\tan x}{\sin x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30548a74b41df9c26d29d3d47575a1a95215db0f)
![{\displaystyle 1<{\frac {x}{\sin x}}<{\frac {1}{\cos x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ebb76a615245ad520adebe91315d2fc7ad491cf)
![{\displaystyle \lim _{x\to 0}{\frac {1}{\cos x}}={\frac {1}{1}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c456b9bd4482bde4491d94ffbd43951c04e225)
![{\displaystyle \lim _{x\to 0}{\frac {x}{\sin x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12c729f71684e42916753d281bf9fb60d8510016)
![{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f2fb52f5211c7b7aa69d9e75195afaab5b9d5b1)
![{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f654e25c3bbbbb7825d619fd3526371fe1cb65b2)
A short way to write the limit
is
.
A short way to write the limit
is
.
A short way to write the limit
is
.