limit examples

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

$$\underset{x\to 0}{lim}\frac{\tan x}{x}=\underset{x\to 0}{lim}\left(\frac{\sin x}{x}\cdot \frac{1}{\cos x}\right)=\underset{x\to 0}{lim}\frac{\sin x}{x}\cdot \underset{x\to 0}{lim}\frac{1}{\cos x}.$$

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 $\underset{x\to 0}{lim}{\displaystyle \frac{\sin ax}{\sin bx}}$.  — As above, we can write

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

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

$$\underset{n\to \mathrm{\infty }}{lim}2n\tan \frac{\pi }{n}=\underset{n\to \mathrm{\infty }}{lim}2\pi \frac{\tan \frac{\pi }{n}}{\frac{\pi }{n}}=2\pi \cdot 1=2\pi ,$$

which is the circumference of the circle.

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

$$\arcsin x:=y,$$

the monotonicity of the arcus sine (http://planetmath.org/CyclometricFunctions) function on  $[-1,\mathrm{\hspace{0.17em}1}]$  implies that “$x\to 0$” is equivalent (http://planetmath.org/Equivalent3) to “$y\to 0$”.  Then we can calculate:

$$\underset{x\to 0}{lim}\frac{\arcsin x}{x}=\underset{y\to 0}{lim}\frac{y}{\sin y}=\underset{y\to 0}{lim}\frac{1}{\frac{\sin y}{y}}=\frac{1}{1}=1.$$

Example 5.  One may use the definition of derivative in

$$\underset{x\to 0}{lim}\frac{\arctan x}{x}=\underset{x\to 0}{lim}\frac{\arctan x-\arctan 0}{x-0}={\left[\frac{d}{dx}\arctan x\right]}_{x=0}=\frac{1}{1+0^2}=1.$$