Evaluate the following limit: _ x 0 (a+x)+ (a-x)-2 x - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sin(\text{a}+\text{x})+\sin(\text{a}-\text{x})-2\sin\text{a}}{\text{x}\sin\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin(\text{a}+\text{x})+\sin(\text{a}-\text{x})-2\sin\text{a}}{\text{x}\sin\text{x}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\big(\frac{\text{a}+\text{x}+\text{a}-\text{x}}{2}\big)\cos\big(\frac{\text{a}+\text{x}-\text{a}+\text{x}}{2}\big)-\sin\text{a}}{\text{x}\sin\text{x}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{a}(\cos\text{x}-1)}{\text{x}\sin\text{x}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{-2\sin\text{a}(1-\cos\text{x})}{\text{x}\sin\text{x}}$ $=-2\sin\text{a}\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{a}\frac{\text{x}}{2}}{\big(2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}\big)\text{x}}$ $=-2\sin\text{a}\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{a}\frac{\text{x}}{2}}{\big(\cos\frac{\text{x}}{2}\big)\text{x}}$ $=-2\sin\text{a}\lim\limits_{\text{x}\rightarrow0}\frac{\tan\frac{\text{x}}{2}}{\frac{\text{x}}{2}}\times\frac12$ $=-2\sin\text{a}\times1\times\frac12$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}=1\Big]$ $=-\sin\text{a}$

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