Evaluate the following limit: _ x 0 3x- 5x x^2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos5\text{x}}{\text{x}^2}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos5\text{x}}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\Big(-2\sin\big(\frac{3\text{x}+5\text{x}}{2}\big)\sin\big(\frac{3\text{x}-5\text{x}}{2}\big)\Big)}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{-2\sin4\text{x}\sin(-\text{x})}{\text{x}^2}\Big)$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin4\text{x}\sin\text{x}}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin4\text{x}}{\text{x}}\times\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}$
$=2\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin4\text{x}}{4\text{x}}\times4\Big)\times\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}\Big)$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$
$=8$

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