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Question: \[\lim_{x \rightarrow \pi/2}\frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x} =\]...

lim⁡x→π/2acot⁡x−acos⁡xcot⁡x−cos⁡x=\lim_{x \rightarrow \pi/2}\frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x} =

A

log⁡a\log a

B

log⁡2\log 2

C

a

D

log x

Answer

log⁡a\log a

Explanation

Solution

limx→π/2(acot⁡x−acos⁡xcot⁡x−cos⁡x)=limx→π/2acos⁡x(acot⁡x−cos⁡x−1cot⁡x−cos⁡x)\underset{x \rightarrow \pi/2}{\text{lim}}{}\left( \frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x} \right) = \underset{x \rightarrow \pi/2}{\text{lim}}a^{\cos x}\left( \frac{a^{\cot x - \cos x} - 1}{\cot x - \cos x} \right)

=acos⁡(π/2)limx→π/2(acot⁡x−cos⁡x−1cot⁡x−cos⁡x)=1log⁡a=log⁡a= a^{\cos(\pi/2)}\underset{x \rightarrow \pi/2}{\text{lim}}\left( \frac{a^{\cot x - \cos x} - 1}{\cot x - \cos x} \right) = 1\log a = \log a.