Let k . If lim x 0 + ( sin ( sin k x ) + cos x + x ) 2 x = e 6 , then the value of k is [2024]

Question

Let $k \in ℝ$ . If $\lim_{x \to 0^{+}}$ $(\sin {(\sin k x) + \cos x + x)}^{\frac{2}{x}}$ $= e^{6},$ then the value of $k$ is [2024]

Smart Achievers · Accepted Answer

(2)

Let, $ℓ = \lim_{x \to 0^{+}}$ $(\sin {(\sin k x) + \cos x + x)}^{\frac{2}{x}}$ $= e^{6},$

Taking log on both sides,

$\Rightarrow \ln ℓ = \lim_{x \to 0^{+}} \frac{2}{x} (\sin (\sin k x) + \cos x + x - 1)$

$\Rightarrow \ln ℓ = \lim_{x \to 0^{+}} 2 (\frac{\sin (\sin k x)}{\sin k x} \cdot \frac{\sin k x}{k x} \cdot \frac{k x}{x} + 1 - (\frac{1 - \cos x}{x^{2}}) x)$

$\Rightarrow \ln ℓ = 2 (k + 1)$ $\Rightarrow ℓ = e^{2 (k + 1)} = e^{6}$

$k + 1 = 3 \Rightarrow k = 2$

Solution

Q.
Let $k \in ℝ$ . If $\lim_{x \to 0^{+}}$ $(\sin {(\sin k x) + \cos x + x)}^{\frac{2}{x}}$ $= e^{6},$ then the value of $k$ is [2024]

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Solution

Q. Let k∈ℝ. If limx→0+(sin(sinkx)+cosx+x)2x=e6, then the value of k is [2024]

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Ans. (2) Let, ℓ=limx→0+(sin(sinkx)+cosx+x)2x=e6 Taking log on both sides, ⇒lnℓ=limx→0+2x(sin(sinkx)+cosx+x-1) ⇒lnℓ=limx→0+2(sin(sinkx)sinkx·sinkxkx·kxx+1-(1-cosxx2)x) ⇒lnℓ=2(k+1)⇒ℓ=e2(k+1)=e6 k+1=3⇒k=2

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Q.
Let $k \in ℝ$ . If $\lim_{x \to 0^{+}}$ $(\sin {(\sin k x) + \cos x + x)}^{\frac{2}{x}}$ $= e^{6},$ then the value of $k$ is [2024]