Question

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

• A. $1$
• B. $\frac{\pi }{2}$
• C. $\frac{1}{16}$
• D. $0$

Answer 1

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

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

Let $x=\frac{\pi }{2}+t$

If $x\to \frac{\pi }{2},t\to 0$

$\underset{t\to 0}{lim}\frac{\sin t-\tan t}{-8t^3}$

$=\underset{t\to 0}{lim}\frac{(t-\frac{t^3}{3!}+\frac{t^5}{5!}+\dots )-(t+\frac{t^3}{3}+\frac{2t^5}{15}+\dots )}{-8t^3}$

$=\frac{1}{16}$

We can put $x=\frac{\pi }{2}-t$ and we'll get L.H.L also same.
Since L.H.L = R.H.L the limit exists and is $=\frac{1}{16}$