Prove this limit - exponentials

Question

$$\lim_{x\rightarrow 0} \frac{4^{-1/x}}{3^{-1/x}+5^{-1/x}}$$

L'Hôpital doesn't work...

I know the limit is zero.

Answer 1

hint: Put $y = \dfrac{1}{x}$ and consider $y \to -\infty$, and $y \to +\infty$ separately. Indeed, $4^{-y} = \dfrac{1}{4^y}$, etc.. So: $L_{+} = \displaystyle \lim_{y\to +\infty} \dfrac{\dfrac{1}{4^y}}{\dfrac{1}{3^y}+\dfrac{1}{5^y}}$. Can you continue at this point?