mathop {lim }limits_{x to pi /2} frac{{{a^{co | vedclass

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Question

(A) Let $L = \mathop {\lim }\limits_{x 	o \pi /2} \frac{{{a^{\cot x}} - {a^{\cos x}}}}{{\cot x - \cos x}}$.We can factor out ${a^{\cos x}}$ from the

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$\log a$

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(A) Let $L = \mathop {\lim }\limits_{x 	o \pi /2} \frac{{{a^{\cot x}} - {a^{\cos x}}}}{{\cot x - \cos x}}$.We can factor out ${a^{\cos x}}$ from the numerator:$L = \mathop {\lim }\limits_{x 	o \pi /2} \frac{{{a^{\cos x}}({a^{\cot x - \cos x}} - 1)}}{{\cot x - \cos x}}$.As $x 	o \pi /2$,$\cos x 	o 0$,so ${a^{\cos x}} 	o {a^0} = 1$.Using the standard limit $\mathop {\lim }\limits_{u 	o 0} \frac{{{a^u} - 1}}{u} = \log a$,where $u = \cot x - \cos x$ (note that as $x 	o \pi /2$,$u 	o 0 - 0 = 0$):$L = 1 	imes \log a = \log a$.

$\mathop {\lim }\limits_{x \to \pi /2} \frac{{{a^{\cot x}} - {a^{\cos x}}}}{{\cot x - \cos x}} = $

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