Let k in mathbb R . If lim x rightarrow 0 + (sin (sin k x)+cos x+x) 2/x =e 6 , then the value of k is

Mathematics · Limits, Continuity and Differentiability

JEE Advanced 2024 — Paper 2 — Question 3

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

$\lim _{x \rightarrow 0^{+}}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^{6}$

(it is $1^{\infty}$ form) $\Rightarrow e^{2 \lim _{x \rightarrow 0^{+}} \frac{\sin (\sin k x)+\cos x+x-1}{x}}=e^{6}$

$=e^{2 \lim _{x \rightarrow 0^{+}} \frac{\cos (\sin k x) \times k \cos k x-\sin x+1}{1}}=e^{6}$

(Using L.H. Rule) $\Rightarrow \mathrm{e}^{2(k+1)}=\mathrm{e}^{6}$

$\Rightarrow \mathrm{k}=2$

Answer key and solution verified before publishing.

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