$\lim_{x\to\pi/2} \cos x\cdot\cos(\tan x)$

Question

I wanted to solve the problem without L'Hopital's rule and was having problems in how to show that the limit does not exist.

Any pointers on how I could approach the problem would be super helpful.

Answer 1

To elaborate on Kavi Rama Murthy's comment, using squeeze theorem, since the inequality $$ 0 \le |\cos x \cos(\tan x) | \le | \cos x|$$ holds, we must have desired limit to be equal to 0.