Let k ∈ ℝ . If _ x arrow 0^+ ( ( k x)+ x+x)^ (2)/(x) =e^6 , then the value of k is

Correct answer: B. 2 aligned & l= _ x arrow 0^ + ( ( k x)+ x+x)^ (2)/(x) =e^6 & l= _ x arrow 0^ + (2)/(x)( ( k x)+ x+x-1) & l= _ x arrow 0^ + 2(( ( k x))/( k x) · ( k x)/(k x) · (k x)/(x)+1-((1- x))/(x^2) · x) & l=2(k+1) & l=e^ 2(k+1) =e^6 & k+1=3 & k=2 aligned )

JEE Advanced 2024Paper 2

Exam: JEE Advanced
Year: 2024
Paper: Paper 2
Subject: Mathematics
Chapter: Limits

JEE Advanced 2024 · Paper 2 · Mathematics · Limits

Let k ∈ ℝ . If _ x arrow 0^+ ( ( k x)+ x+x)^ (2)/(x) =e^6 , then...

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

Options

A $1$
B $2$
C $3$
D $4$

Correct answer

B. $2$

Step-by-step solution

\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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