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Q)

The value of $\lim\limits_{x\to 0}\big((\sin x)^{1/x}+(1+x)^{\large\sin x}\big)$ where $x > 0$ is

$(a)\;0\qquad(b)\;-1\qquad(c)\;1\qquad(d)\;2$

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A)
$\lim\limits_{x\to 0}\big[(\sin x)^{1/x}+(1/x)^{\large\sin x}\big]$
$\Rightarrow \lim\limits_{x\to 0}(\sin x)^{1/x}+\lim\limits_{x\to 0}\big(\large\frac{1}{x}\big)^{\large \sin x}$
$\Rightarrow 0+e^{\large\lim\limits_{x\to 0}\sin x\log(1/x)}$
$\mid\sin x\mid < 1\;when \;x\rightarrow 0]$
$\Rightarrow e^{\large\lim\limits_{x\to 0}\Large\frac{-\log x}{cosec x}}$
$\Rightarrow e^{\large\lim\limits_{x\to 0}\Large\frac{-1/x}{-cosec x\cot x}}$
[Using L Hospital rule]
$\Rightarrow e^{\large\lim\limits_{x\to 0}\Large\frac{\sin x}{x}\normalsize\tan x}$
$\Rightarrow e^0=1$
Hence (c) is the correct answer.
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