Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{2\text{x}-\sin\text{x}}{\tan\text{x}+\text{x}}$
$\lim\limits_{\text{x}\rightarrow0}\frac{2\text{x}-\sin\text{x}}{\tan\text{x}+\text{x}}$
Dividing each term by x
$=\lim\limits_{\text{x}\rightarrow0}\frac{2-\frac{\sin\text{x}}{\text{x}}}{\frac{\tan\text{x}}{\text{x}}+1}$
$=\frac{\lim\limits_{\text{x}\rightarrow0}2-\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}}{\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}+\lim\limits_{\text{x}\rightarrow0}+1}$
$=\frac{2-1}{1+1}$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1,\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}=1\Big]$
$=\frac{1}{2}$
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