Question

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

• A. $\frac{1}{24}$
• B. $\frac{1}{16}$
• C. $\frac{1}{8}$
• D. $\frac{1}{4}$

Answer 1

The correct option is B $\frac{1}{16}$

Put $x=\frac{\pi }{2}-h$
As $x\to \frac{\pi }{2},h\to 0$
$\Rightarrow \underset{x\to \frac{\pi }{2}}{lim}\frac{\tan h-\sin h}{(2h{)}^3}$

$=\underset{h\to 0}{lim}\frac{\frac{\sin h}{\cos h}-\sin h}{8h^3}$

$=\underset{h\to 0}{lim}\frac{\frac{\sin h-\cos h\sin h}{\cos h}}{8h^3}$

$=\underset{h\to 0}{lim}\frac{\sin h(1-\cos h)}{8h^3\cdot \cos h}$

$=\underset{h\to 0}{lim}\frac{1}{8}\frac{\sin h}{h}\cdot \frac{1-\cos h}{h^2}\cdot \frac{1}{\cos h}=\frac{1}{8}\cdot 1\cdot \frac{1}{2}\cdot 1=\frac{1}{16}$