limit rules of functionsLimitRulesOfFunctions2004-11-25 12:25:202008-01-26 15:22:00Theoremlimit% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem{thmplain}{Theorem}\begin{thmplain}
\, Let $f$ and $g$ be two \PMlinkname{real}{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
\begin{enumerate}
\item $\lim_{x\to x_0}[f(x)\!\pm\!g(x)]
= \lim_{x\to x_0}f(x)\pm\lim_{x\to x_0}g(x),$
\item $\lim_{x\to x_0}f(x)g(x) = \lim_{x\to x_0}f(x)\cdot\lim_{x\to x_0}g(x),$
\item $\lim_{x\to x_0}\frac{f(x)}{g(x)} =
\frac{\lim_{x\to x_0}f(x)}{\lim_{x\to x_0}g(x)},$
\item $\lim_{x\to x_0}c = c
\quad\mathrm{where}\,\,c\,\,\mathrm{is\,\,a\,\,constant}.$
\end{enumerate}
\end{thmplain}
These rules are used in limit calculations and in proving the corresponding differentiation rules (sum rule, product rule etc.).
In \PMlinkescapetext{theorem} 1, the domains of $f$ and $g$ could be any topological space (not necessarily $\mathbb{R}$ or $\mathbb{C}$).
There are similar limit rules of sequences.
As well, one often needs the
\begin{thmplain}
\,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
$$\lim_{x\to x_0}g(f(x)) = g(\lim_{x\to x_0}f(x)).$$
\end{thmplain}