_ x 0 x x = _ x 0 x x 1 x = _ x 0 ( x x ) _ x 0 ( 1 x ) [ _ x a f(x)g(x) = _ x a f(x) _ x a g(x) ] = 1 1 0 = 1 1 1 = 1

Evaluate _ x 0 x x

Question

Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{x}$

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Answer

$\mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{x} = \mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} \cdot \frac{1}{{\cos x}}$
$ = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\sin x}}{x}} \right) \cdot \mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{{\cos x}}} \right)$
$\left[\because {\mathop {\lim }\limits_{x \to a} f(x)g(x) = \mathop {\lim }\limits_{x \to a} f(x)\mathop {\lim }\limits_{x \to a} g(x)} \right]$
$= 1 \times \frac { 1 } { \cos 0 } = 1 \times \frac { 1 } { 1 } = 1$

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