Evaluate the following limit: _ x 0 1+ - 1- x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\sin\text{x}}-\sqrt{1-\sin\text{x}}}{\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\sin\text{x}}-\sqrt{1-\sin\text{x}}}{\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\big(\sqrt{1+\sin\text{x}}-\sqrt{1-\sin\text{x}}\big)}{\text{x}}\frac{\big(\sqrt{1+\sin\text{x}}+\sqrt{1-\sin\text{x}}}{\big(\sqrt{1+\sin\text{x}}+\sqrt{1-\sin\text{x}}\big)}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\big((1+\sin\text{x})+(1-\sin\text{x})\big)}{\text{x}\big(\sqrt{1+\sin\text{x}}+\sqrt{1-\sin\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}}{\text{x}\big(\sqrt{1+\sin\text{x}}+\sqrt{1-\sin\text{x}}\big)}$
$=2\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}\times\frac{1}{\lim\limits_{\text{x}\rightarrow0}\big(\sqrt{1+\sin\text{x}}+\sqrt{1-\sin\text{x}}\big)}$
$=2\times1\times\frac12$
$=1$

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