If limx→0 [cot⁡(π/4+x)]1/x=Ae2 then the value of A is

Ae2=limx→0 cot⁡x−1cot⁡x+11x=limx→0 1−2cot⁡x+1−cot⁡x+12×−2tan⁡x1+tan⁡x=e−2

If limx→0 [cot⁡(π/4+x)]1/x=Ae2 then the value of A is

Ae2=limx→0 cot⁡x−1cot⁡x+11x=limx→0 1−2cot⁡x+1−cot⁡x+12×−2tan⁡x1+tan⁡x=e−2

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If $lim_{x \to 0} [\cot (π / 4 + x)]^{1 / x} = A e^{2}$ then the value of $A$ is

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$e^{2}$

$e^{- 4}$

$e^{}$

$e^{3}$

answer is B.

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Detailed Solution

$A e^{2} = lim_{x \to 0} {(\frac{\cot x - 1}{\cot x + 1})}^{\frac{1}{x}}$

$= lim_{x \to 0} {(1 - \frac{2}{\cot x + 1})}^{- \frac{\cot x + 1}{2} \times \frac{- 2 \tan x}{1 + \tan x}} = e^{- 2}$

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If limx→0 [cot⁡(π/4+x)]1/x=Ae2 then the value of A is

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If $lim_{x \to 0} [\cot (π / 4 + x)]^{1 / x} = A e^{2}$ then the value of $A$ is