How does this function become indeterminate?

Question

This is just a quick question, that will probably be easily solved by you calculus wizards out there.

$$ \lim_{x\to\frac{1}{2}} f(x) =\frac{x{\cos {\left(\pi{x}\right)}}}{e^{x}-\sqrt{e}}=\frac{\approx 0.49}{0}$$

I thought indeterminate form was found by making $x = 1/2$ and then seeing if $f(x) = 0$ and $g(x)=0$, which here only seems to be true for the function on the bottom.

Oh, and i know the value is closing up to zero over zero, I'm just asking how.

Answer 1

Please use L'Hôpital's rule https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule

So in order to get the limit, you should do calculate the limit for the derivative of the function first, then calculate the limit. The answer is \frac{-1/2 \pi}{\sqrt(e)}