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$\lim _{x \rightarrow 0}\left(\frac{1+\tan x}{1+\sin x}\right)^{\operatorname{cosec} x}$ is equal to
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$1$
$\begin{aligned} & \text { Given limit }=\lim _{x \rightarrow 0}\left[(1+\tan x)^{\operatorname{cosec} x} \times 1 /(1+\sin x)^{\operatorname{cosec} x}\right] \\ & \left.=\lim _{x \rightarrow 0}\left[\{1+\tan x)^{\cot x}\right\}^{\sec x} \times\left\{1 /(1+\sin x)^{\operatorname{cosec} x}\right\}\right] \\ & =e^{\sec (0)} \cdot \frac{1}{e}=e \cdot \frac{1}{e}=1\end{aligned}$
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