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26A06-LimitOfdisplaystylefrac1cosXxAsXApproaches0.tex
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26A06-LimitOfdisplaystylefrac1cosXxAsXApproaches0.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{LimitOfdisplaystylefrac1cosXxAsXApproaches0}
\pmcreated{2013-03-22 16:58:49}
\pmmodified{2013-03-22 16:58:49}
\pmowner{Wkbj79}{1863}
\pmmodifier{Wkbj79}{1863}
\pmtitle{limit of $\displaystyle \frac{1-\cos x}{x}$ as $x$ approaches 0}
\pmrecord{7}{39256}
\pmprivacy{1}
\pmauthor{Wkbj79}{1863}
\pmtype{Corollary}
\pmcomment{trigger rebuild}
\pmclassification{msc}{26A06}
\pmclassification{msc}{26A03}
\pmrelated{DerivativesOfSinXAndCosX}
\endmetadata
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{psfrag}
\usepackage{graphicx}
\usepackage{amsthm}
%%\usepackage{xypic}
\newtheorem{cor*}{Corollary}
\begin{document}
\begin{cor*}
\[
\lim_{x \to 0} \frac{1-\cos x}{x}=0
\]
\end{cor*}
\begin{proof}
\begin{center}
\begin{tabular}{rl}
$\displaystyle \lim_{x \to 0} \frac{1-\cos x}{x}$ & $=\displaystyle \lim_{x \to 0} \frac{(1-\cos x)(1+\cos x)}{x(1+\cos x)}$ \\
& \\
& $=\displaystyle \lim_{x \to 0} \frac{1-\cos x+\cos x-\cos^2 x}{x(1+\cos x)}$ \\
& \\
& $=\displaystyle \lim_{x \to 0} \frac{1-\cos^2 x}{x(1+\cos x)}$ \\
& \\
& $=\displaystyle \lim_{x \to 0} \frac{\sin^2 x}{x(1+\cos x)}$ \\
& \\
& $=\displaystyle \lim_{x \to 0} \frac{\sin x}{x} \cdot \frac{\sin x}{1+\cos x}$ \\
& \\
& $=\displaystyle \left( \lim_{x \to 0} \frac{\sin x}{x} \right) \left( \lim_{x \to 0} \frac{\sin x}{1+\cos x} \right)$ by \PMlinkname{this entry}{LimitRulesOfFunctions} \\
& \\
& $=\displaystyle 1 \cdot \lim_{x \to 0} \frac{\sin x}{1+\cos x}$ by \PMlinkname{this entry}{LimitOfDisplaystyleFracsinXxAsXApproaches0} \\
& \\
& $=\displaystyle \frac{\sin 0}{1+\cos 0}$ by \PMlinkname{this entry}{LimitRulesOfFunctions} and the fact that $\sin$ and $\cos$ are continuous \\
& \\
& $=\displaystyle \frac{0}{1+1}$ \\
& \\
& $=0$ \qedhere
\end{tabular}
\end{center}
\end{proof}
This corollary has an obvious corollary to it:
\begin{cor*}
\[
\lim_{x \to 0} \frac{\cos x-1}{x}=0
\]
\end{cor*}
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\end{document}