Evaluate the following limit: _ x 0 3x- - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}-\sin\text{x}}{\sin\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}-\sin\text{x}}{\sin\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg(\frac{2\cos\big(\frac{3\text{x}+\text{x}}{2}\big)\sin\big(\frac{3\text{x}-\text{x}}{2}\big)}{\sin\text{x}}\Bigg)$
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{2\cos2\text{x}\sin\text{x}}{\sin\text{x}}\Big)$
$=\lim\limits_{\text{x}\rightarrow0}(2\cos\text{x})$
$=2\lim\limits_{\text{x}\rightarrow0}\cos2\text{x}$
$=2\times\cos0$
$=2\times1=2$
$=2$

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