Question

Solve $\underset{x\to \frac{\pi }{2}}{lim}\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cos x}a>0$

• A. ${\log }_e\frac{\pi }{2}$
• B. ${\log }_{e}2$
• C. ${\log }_{e}a$
• D. $a$

Answer 1

Answer: (C)

${\log }_{e}a$

$\underset{x\to \frac{\pi }{2}}{lim}\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cos x}$

$=\underset{x\to \frac{\pi }{2}}{lim}\frac{[1+\cot x{log}_{e}a+\frac{{cot}^{2}x}{2!}{\left({log}_{e}a\right)}^2+...-1-\cos x{log}_{e}a-\frac{{cos}^{2}x}{2!}{\left({log}_{e}a\right)}^2-...]}{\cot x-\cos x}$

$=\underset{x\to \frac{\pi }{2}}{lim}\frac{(\cot x-\cos x)}{\cot x-\cos x}$

$=\underset{x\to \frac{\pi }{2}}{lim}\{{log}_{e}a+\frac{\cot x+\cos x}{2!}{\left({log}_{e}a\right)}^2+...\}$

$={log}_{e}a$