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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}(\cos\text{x}+\sin\text{x})^{\frac{1}{\text{x}}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}(\cos\text{x}+\sin\text{x})^{\frac{1}{\text{x}}}$
$=\ \lim\limits_{\text{x}\rightarrow0}(1+(\cos\text{x}+\sin\text{x}-1))^{\frac{1}{\text{x}}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{(\cos\text{x}+\sin\text{x}-1)}{\text{x}}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{(\sin\text{x}-(1-\cos\text{x}))}{\text{x}}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sin\text{x}-2\sin^2\big(\frac{\text{x}}{2}\big)\big)}{\text{x}}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}-\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\big(\frac{\text{x}}{2}\big)\sin\big(\frac{\text{x}}{2}\big)}{2\big(\frac{\text{x}}{2}\big)}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}-\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\big(\frac{\text{x}}{2}\big)\sin\big(\frac{\text{x}}{2}\big)}{\big(\frac{\text{x}}{2}\big)}}$
$=\text{e}^{1-0}$
$=\text{e}$

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