Which expressions $ f(\frac{\pi}2)=0 $ or $\lim_{x\to \frac{\pi}2} f(x)=0 $ is correct for $f(x)=\frac{2\cos x}{4+5\tan x}$?

Question

I have this function

$$ f(x)=\frac{2\cos x}{4+5\tan x} , $$

and I want to calculate the value of the function at $x=\frac{\pi}2$, the result is $\frac{0}{\infty}$; then, which expression should I use?

$$ f(\frac{\pi}2)=0 $$

$$ \lim_{x\to \frac{\pi}2} f(x)=0 $$

Answer 1

First of all, the function is undefined at $x = \frac{\pi}{2}$. So you can't find the value of $f(\frac{\pi}{2})$. But, if you are considering the limit, $$\lim_{x \to \frac{\pi}{2}} \frac{2 \cos x}{4+5 \tan x}.$$ Because $\tan x = \frac{\sin x}{\cos x}$, we get $$\lim_{x \to \frac{\pi}{2}} \frac{2 \cos x}{\frac{4 \cos x+5 \sin x}{\cos x}}$$ $$ \iff \lim_{x \to \frac{\pi}{2}} \frac{2 \cos^2 x}{4 \cos x+5 \sin x} =0.$$

We obtain the value of its limit is $= 0$.

Even the form of its limit is $\frac{0}{\infty}$, the limit goes to $0.$ This is not indeterminate forms.