Question

$\underset{x\to \frac{\pi }{2}}{lim}\frac{\cot x-\cos x}{(\frac{\pi }{2}-x{)}^3}$=

• A. $\frac{-1}{2}$
• B. $\frac{1}{2}$
• C. $2$
• D. $-2$

Answer 1

The correct option is B $\frac{1}{2}$

$L=\underset{x\to \frac{\pi }{2}}{lim}\frac{\cot x-\cos x}{(\frac{\pi }{2}-x{)}^3}$
Put $h=\frac{\pi }{2}-x$
$L=\underset{h\to 0}{lim}\frac{\tan h-\sin h}{h^3}=\underset{h\to 0}{lim}\frac{h+\frac{h^3}{3}-(h-\frac{h^3}{3!})}{h^3}$ $=\frac{1}{3}+\frac{1}{6}=\frac{1}{2}$