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. 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. 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. 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. 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