Evaluate the following limit: _ 0 4 3 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\theta\rightarrow0}\frac{\sin4\theta}{\tan3\theta}$

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Answer

$\lim\limits_{\theta\rightarrow0}\frac{\sin4\theta}{\tan3\theta}$
$=\frac{\lim\limits_{\theta\rightarrow0}\sin4\theta}{\lim\limits_{\theta\rightarrow0}\tan3\theta}$
$=\frac{\Big(\lim\limits_{\theta\rightarrow0}\frac{\sin4\theta}{4\theta}\Big)\times4\theta}{\Big(\lim\limits_{\theta\rightarrow0}\frac{\tan3\theta}{3\theta}\Big)\times3\theta}$
$=\frac{1\times4\theta}{1\times3\theta}$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$=\frac43$

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