Let $k \in \mathbb{R}$. If $\lim _{x \rightarrow 0^+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6$, then the value of $k$ is

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Aicharya · Accepted Answer

$\begin{aligned} & l=\lim _{x \rightarrow 0^{+}}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6 \ & \Rightarrow \ln l=\lim _{x \rightarrow 0^{+}} \frac{2}{x}(\sin (\sin k x)+\cos x+x-1) \ & \Rightarrow \ln l=\lim _{x \rightarrow 0^{+}} 2\left(\frac{\sin (\sin k x)}{\sin k x} \cdot \frac{\sin k x}{k x} \cdot \frac{k x}{x}+1-\frac{(1-\cos x)}{x^2} \cdot x\right) \ & \Rightarrow \ln l=2(k+1) \ & \Rightarrow l=e^{2(k+1)}=e^6 \ & k+1=3 \ & \Rightarrow k=2\end{aligned}$

Let $k \in \mathbb{R}$. If $\lim _{x \rightarrow 0^+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6$, then the…

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