Evaluate the following limit: _ x 0 ax+x b - Vidyadip

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{ax}+\text{x}\cos\text{x}}{\text{b}\sin\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}+\cos\text{x}}{\frac{\text{b}\sin\text{x}}{\text{x}}}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}\text{a}+\lim\limits_{\text{x} \rightarrow0}\cos\text{x}}{\lim\limits_{\text{x}\rightarrow0}\frac{\text{b}\sin\text{x}}{\text{x}}}$
$=\frac{\text{a}+1}{\text{b}}$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$=\frac{\text{a}+1}{\text{b}}$

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