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 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 factor is 1 by the parent entry. 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 $$\displaystyle\lim_{x\to 0}\frac{\sin{ax}}{\sin{bx}} = \lim_{x\to0}\left(\frac{\sin{ax}}{ax}\cdot\frac{bx}{\sin{bx}}\cdot\frac{a}{b}\right) = \lim_{x\to0}\frac{\sin{ax}}{ax}\cdot\lim_{x\to0}\frac{bx}{\sin{bx}}\cdot\lim_{x\to0}\frac{a}{b} = 1\cdot1\cdot\frac{a}{b} = \frac{a}{b}.$$

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.

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 function on $[-1,\,1]$ implies that ``$x\to0$ '' is equivalent to ``$y\to0$ ''. Then we can calculate: $$\lim_{x\to 0}\frac{\arcsin{x}}{x} = \lim_{y\to0}\frac{y}{\sin{y}} = \lim_{y\to0}\frac{1}{\frac{\sin{y}}{y}} = \frac{1}{1} = 1.$$

Example 5. One may use the definition of derivative in $$\lim_{x\to0}\frac{\arctan{x}}{x} = \lim_{x\to0}\frac{\arctan{x}-\arctan{0}}{x-0} = \left[\frac{d}{dx}\arctan{x}\right]_{x=0} = \frac{1}{1+0^2} = 1.$$