Evaluate the following limit: _ 0 3 2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\theta\rightarrow0}\frac{\sin3\theta}{\tan2\theta}$

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Answer

$\lim\limits_{\theta\rightarrow0}\frac{\sin3\theta}{\tan2\theta}$ $=\frac{\lim\limits_{\theta\rightarrow0}\sin3\theta}{\lim\limits_{\theta\rightarrow0}\tan2\theta}$ $=\frac{\lim\limits_{3\theta\rightarrow0}\frac{\sin3\theta}{3\theta}\times3\theta}{\lim\limits_{2\theta\rightarrow0}\frac{\tan2\theta}{2\theta}\times2\theta}$ $=\frac{\Big(\lim\limits_{3\theta\rightarrow0}\frac{\sin3\theta}{3\theta}\Big)}{\Big(\lim\limits_{2\theta\rightarrow0}\frac{\tan2\theta}{2\theta}\Big)}\times\frac{3\theta}{2\theta}$$=\frac11\times\frac32$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\text{ and }\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}=1\Big]$
$=\frac32$

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