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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}\cos\text{x}}{3\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}\cos\text{x}}{3\text{x}}$
$=\frac13\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}\cos\text{x}}{{\text{x}}}{}$
$=\frac13\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{{\text{x}}}\times\lim\limits_{\text{x}\rightarrow0}\cos\text{x}$
$=\frac13\times1\times1$ $\Big[\because\ \lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1,\lim\limits_{\text{x}\rightarrow0}\cos0^\circ=1\Big]$
$=\frac13$

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