Evaluate the following limit: _ x 0 (3+x)- (3-x) x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin(3+\text{x})-\sin(3-\text{x})}{\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin(3+\text{x})-\sin(3-\text{x})}{\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\cos\big(\frac{3+\text{x}+3-\text{x}}{2}\big)\sin\big(\frac{3+\text{x}-3+\text{x}}{2}\big)}{\text{x}}$
$=2\lim\limits_{\text{x} \rightarrow0}\frac{\cos3.\sin\text{x}}{\text{x}}$
$=2\cos3\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}$
$=2\cos3\times1$
$=2\cos3$

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