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'limit rules of functions'
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| Title of object: |
limit rules of functions |
| Canonical Name: |
LimitRulesOfFunctions |
| Type: |
Theorem |
| Created on: |
2004-11-25 12:25:20 |
| Modified on: |
2005-04-11 02:45:57 |
| Classification: |
msc:26A06, msc:30A99 |
Revision comment (for changes between this and next version):
Changes for correction #8590 ('wording/grammar').
Thank you very much. |
Preamble:
% 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} |
Content:
\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 the \PMlinkescapetext{Theorem} 1, the \PMlinkname{domains}{Function} of $f$ and $g$ could instead $\mathbb{R}$ or $\mathbb{C}$ be any topological space.
There are similar limit rules of sequences.
As well, one needs often the
\begin{thmplain}
\,If there exists the limit \,$\lim_{x\to x_0}f(x) = a$\, and if $g$ is continuous in 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} |
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