First slide

Introduction to limits

Question

$lim_{x \to π / 2} \frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x}, a > 0$ is equal to

Moderate

A

$\log_{e} (\frac{π}{2})$
B

$\log_{e} 2$
C

$\log_{e} a$
D

$a$

Solution

We have,
$\begin{array}{l} = lim_{=\to - / 2} \frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x} \\ = lim_{=\to π / 2} a^{\cos x} (\frac{a^{\cot x - \cos x} - 1}{\cot x - \cos x}) = a^{\cos π / 2} \times \log_{e} a = \log_{e} a \end{array}$

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