# limit examples

Example 1.  Determine the limit $\displaystyle\lim_{x\to 0}\frac{\tan{x}}{x}$.  — Using the definition of $\tan$ and the limit rule of product (http://planetmath.org/LimitRulesOfFunctions) we can write

 $\displaystyle\lim_{x\to 0}\frac{\tan{x}}{x}=\lim_{x\to 0}\left(\frac{\sin{x}}{% x}\cdot\frac{1}{\cos{x}}\right)=\lim_{x\to 0}\frac{\sin{x}}{x}\cdot\lim_{x\to 0% }\frac{1}{\cos{x}}.$

The limit in the former factor (http://planetmath.org/Product) is 1 by the parent entry (http://planetmath.org/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 $1$.

Example 2.  Determine the limit $\displaystyle\lim_{x\to 0}\frac{\sin{ax}}{\sin{bx}}$.  — As above, we can write

 $\displaystyle\lim_{x\to 0}\frac{\sin{ax}}{\sin{bx}}=\lim_{x\to 0}\left(\frac{% \sin{ax}}{ax}\cdot\frac{bx}{\sin{bx}}\cdot\frac{a}{b}\right)=\lim_{x\to 0}% \frac{\sin{ax}}{ax}\cdot\lim_{x\to 0}\frac{bx}{\sin{bx}}\cdot\lim_{x\to 0}% \frac{a}{b}=1\cdot 1\cdot\frac{a}{b}=\frac{a}{b}.$

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

 $\lim_{n\to\infty}2n\tan\frac{\pi}{n}=\lim_{n\to\infty}2\pi\frac{\tan\frac{\pi}% {n}}{\frac{\pi}{n}}=2\pi\cdot 1=2\pi,$

which is the circumference of the circle.

Example 4.  Determine the limit $\displaystyle\lim_{x\to 0}\frac{\arcsin{x}}{x}$.  — If we denote

 $\arcsin{x}:=y,$

the monotonicity of the arcus sine (http://planetmath.org/CyclometricFunctions) function on  $[-1,\,1]$  implies that “$x\to 0$” is equivalent (http://planetmath.org/Equivalent3) to “$y\to 0$”.  Then we can calculate:

 $\lim_{x\to 0}\frac{\arcsin{x}}{x}=\lim_{y\to 0}\frac{y}{\sin{y}}=\lim_{y\to 0}% \frac{1}{\frac{\sin{y}}{y}}=\frac{1}{1}=1.$

Example 5.  One may use the definition of derivative in

 $\lim_{x\to 0}\frac{\arctan{x}}{x}=\lim_{x\to 0}\frac{\arctan{x}-\arctan{0}}{x-% 0}=\left[\frac{d}{dx}\arctan{x}\right]_{x=0}=\frac{1}{1+0^{2}}=1.$
Title limit examples LimitExamples 2013-03-22 17:40:16 2013-03-22 17:40:16 pahio (2872) pahio (2872) 8 pahio (2872) Example msc 26A06 msc 26A03 utilizing limit of $\frac{\sin{x}}{x}$ in 0 LimitRulesOfFunctions DerivativeOfInverseFunction ListOfCommonLimits