limit examplesLimitExamples2007-12-06 16:37:082008-05-15 13:58:00Examplelimit of $\displaystyle \frac{\sin x}{x}$ as $x$ approaches 0% this is the default PlanetMath preamble. as your knowledge
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\textbf{Example 1.}\, Determine the limit $\displaystyle\lim_{x\to 0}\frac{\tan{x}}{x}$.\; --- Using the definition of $\tan$ and the \PMlinkname{limit rule of product}{LimitRulesOfFunctions} we can write
$$\displaystyle\lim_{x\to 0}\frac{\tan{x}}{x} =
\lim_{x\to0}\left(\frac{\sin{x}}{x}\cdot\frac{1}{\cos{x}}\right) =
\lim_{x\to0}\frac{\sin{x}}{x}\cdot\lim_{x\to0}\frac{1}{\cos{x}}.$$
The limit in the former \PMlinkname{factor}{Product} is 1 by the \PMlinkname{parent entry}{LimitOfDisplaystyleFracsinXxAsXApproaches0}. Also the latter limit is 1, since\, $\cos{x}$ and thus the quotient $\displaystyle\frac{1}{\cos{x}}$ is continuous in the point \,$x = 0$ (see continuity of sine and cosine).\, Accordingly the desired limit is $1$.\\
\textbf{Example 2.}\, Determine the limit $\displaystyle\lim_{x\to 0}\frac{\sin{ax}}{\sin{bx}}$.\; --- As above, we can write
$$\displaystyle\lim_{x\to 0}\frac{\sin{ax}}{\sin{bx}} =
\lim_{x\to0}\left(\frac{\sin{ax}}{ax}\cdot\frac{bx}{\sin{bx}}\cdot\frac{a}{b}\right) =
\lim_{x\to0}\frac{\sin{ax}}{ax}\cdot\lim_{x\to0}\frac{bx}{\sin{bx}}\cdot\lim_{x\to0}\frac{a}{b}
= 1\cdot1\cdot\frac{a}{b} = \frac{a}{b}.$$
\textbf{Example 3.}\, The perimeter of a regular $n$-gon, circumscribed to a circle with radius 1, is $2n\tan\frac{\pi}{n}$.\, Determine the limit of this perimeter as $n$ tends to infinity.\, --- Utilising the example 1 we can calculate
$$\lim_{n\to\infty}2n\tan\frac{\pi}{n} = \lim_{n\to\infty}2\pi\frac{\tan\frac{\pi}{n}}{\frac{\pi}{n}} = 2\pi\cdot1 = 2\pi,$$
which is the circumference of the circle.\\
\textbf{Example 4.}\, Determine the limit $\displaystyle\lim_{x\to 0}\frac{\arcsin{x}}{x}$.\; --- If we denote\, $$\arcsin{x} := y,$$
the monotonicity of the \PMlinkname{arcus sine}{CyclometricFunctions} function on\, $[-1,\,1]$\, implies that ``$x\to0$'' is \PMlinkname{equivalent}{Equivalent3} to ``$y\to0$''.\, Then we can calculate:
$$\lim_{x\to 0}\frac{\arcsin{x}}{x} = \lim_{y\to0}\frac{y}{\sin{y}} = \lim_{y\to0}\frac{1}{\frac{\sin{y}}{y}} = \frac{1}{1} = 1.$$
\textbf{Example 5.}\, One may use the definition of derivative in
$$\lim_{x\to0}\frac{\arctan{x}}{x} = \lim_{x\to0}\frac{\arctan{x}-\arctan{0}}{x-0} = \left[\frac{d}{dx}\arctan{x}\right]_{x=0} = \frac{1}{1+0^2} = 1.$$