JEE Advanced · Mathematics · 16. Limits
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
- A \(1\)
- B \(2\)
- C \(3\)
- D \(4\)
Answer & Solution
Correct Answer
(B) \(2\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & l=\lim _{x \rightarrow 0^{+}}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6 \\ & \Rightarrow \ln l=\lim _{x \rightarrow 0^{+}} \frac{2}{x}(\sin (\sin k x)+\cos x+x-1) \\ & \Rightarrow \ln l=\lim _{x \rightarrow 0^{+}} 2\left(\frac{\sin (\sin k x)}{\sin k x} \cdot \frac{\sin k x}{k x} \cdot \frac{k x}{x}+1-\frac{(1-\cos x)}{x^2} \cdot x\right) \\ & \Rightarrow \ln l=2(k+1) \\ & \Rightarrow l=e^{2(k+1)}=e^6 \\ & k+1=3 \\ & \Rightarrow k=2\end{aligned}\)
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