limit rules of functions

Theorem 1.

Let $f$ and $g$ be two real (http://planetmath.org/RealFunction) or complex functions. Suppose that there exist the limits $\lim_{x\to x_{0}}f(x)$ and $\lim_{x\to x_{0}}g(x)$ . Then there exist the limits $\lim_{x\to x_{0}}[f(x)\!\pm\!g(x)]$ , $\lim_{x\to x_{0}}f(x)g(x)$ and, if $\lim_{x\to x_{0}}g(x)\neq 0$ , also $\lim_{x\to x_{0}}f(x)/g(x)$ , and

1.

$\lim_{x\to x_{0}}[f(x)\!\pm\!g(x)]\;=\;\lim_{x\to x_{0}}f(x)\pm\lim_{x\to x_{0% }}g(x),$
2.

$\lim_{x\to x_{0}}f(x)g(x)\;=\;\lim_{x\to x_{0}}f(x)\cdot\lim_{x\to x_{0}}g(x),$
3.

$\lim_{x\to x_{0}}\frac{f(x)}{g(x)}\;=\;\frac{\lim_{x\to x_{0}}f(x)}{\lim_{x\to x% _{0}}g(x)},$
4.

$\lim_{x\to x_{0}}c\;=\;c\quad\mathrm{where}\,\,c\,\,\mathrm{is\,\,a\,\,% constant}.$

These rules are used in limit calculations and in proving the corresponding differentiation rules (sum rule, product rule etc.).

In 1, the domains of $f$ and $g$ could be any topological space (not necessarily $\mathbb{R}$ or $\mathbb{C}$ ).

There are limit rules of sequences (http://planetmath.org/Sequence).

As well, one often needs the

Theorem 2.

If there exists the limit $\lim_{x\to x_{0}}f(x)=a$ and if $g$ is continuous at the point $x=a$ , then there exists the limit $\lim_{x\to x_{0}}g(f(x))$ , and

\lim_{x\to x_{0}}g(f(x))\;=\;g(\lim_{x\to x_{0}}f(x)).

Title	limit rules of functions
Canonical name	LimitRulesOfFunctions
Date of creation	2013-03-22 14:51:21
Last modified on	2013-03-22 14:51:21
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	20
Author	pahio (2872)
Entry type	Theorem
Classification	msc 26A06
Classification	msc 30A99
Synonym	limit rules of sequences
Related topic	GrowthOfExponentialFunction
Related topic	ImproperLimits
Related topic	DerivativesOfSineAndCosine
Related topic	ListOfCommonLimits
Related topic	LimitExamples
Related topic	ProductAndQuotientOfFunctionsSum
Related topic	DerivationOfPlasticNumber