Evaluate the following limit: _ x 0 ^ x^ - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}^\circ}{\text{x}^\circ}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}^\circ}{\text{x}^\circ}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{{\sin\frac{\text{x}\times\pi}{180}}}{{\text{x}}\times\frac{\pi}{180}}$ $\Big[\because1^\circ=\frac{\pi}{180}\text{ radious}\Big]$
$=\lim\limits_{{\frac{\pi\text{x}}{180}}\rightarrow0}\frac{\sin\frac{\text{x}\times\pi}{180}}{\frac{\pi\text{x}}{180}}$ $\Big[\because\text{if x}\rightarrow0\text{ then }\frac{\pi\text{x}}{180}\rightarrow0\Big]$
$=1$ $\Big[\because\ \lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}=1\Big]$

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