find this limit $\lim_{x+\rightarrow 0}e^{-1/x}/x^2$

Question

I want to find this limit

$$\lim_{x_+\rightarrow 0}\frac{e^{-1/x}}{x^2}$$

I know that exponential convergence rate is faster, but how to find the limit mathematically?

thanks.

Answer 1

$\lim_{x\rightarrow 0^+}\frac{e^{-1/x}}{x^2}=\lim_{x\rightarrow 0^+}\frac{1/x^{2}}{e^{1/x}}=\lim_{x\rightarrow 0^+}\frac{2/x}{e^{1/x}}=\lim_{x\rightarrow 0^+}\frac{-2}{e^{1/x}}=0$ after two applications of L'Hospital.