Evaluate the following limit: _ x 0 5x- 3x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin5\text{x}-\sin3\text{x}}{\sin\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin5\text{x}-\sin3\text{x}}{\sin\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\cos\big(\frac{5\text{x}+3\text{x}}{2}\big)\sin\big(\frac{5\text{x}-3\text{x}}{2}\big)}{\sin\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\cos4\text{x}\sin\text{x}}{\sin\text{x}}$
$=2\lim\limits_{\text{x}\rightarrow0}\cos4\text{x}$
$=2\times\cos0$
$=2$

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