How to solve $\lim _{x\to 0+}\left(\frac{\frac{e^{-\frac{1}{x}}}{x^3} +e^{-\frac{1}{\sqrt{x}}}}{e^{-\frac{1}{x}}lnx}\right)$?

Question

I have a problem with this limit, i have no idea how to compute it. Can you explain the method and the steps used? Thanks $$\lim _{x\to 0+}\left(\frac{\frac{e^{-\frac{1}{x}}}{x^3} +e^{-\frac{1}{\sqrt{x}}}}{e^{-\frac{1}{x}}lnx}\right)$$

Answer 1

It is convenient to make the change of variables $x=1/t$ to get $$ \lim_{t\to +\infty}\frac{t^3 e^{-t}+e^{-\sqrt{t}}}{-e^{-t}\ln t}= \lim_{t\to +\infty}\left[-\frac{t^3}{\ln t}-\frac{e^{t-\sqrt{t}}}{\ln t}\right]\ . $$ The term $t^3/\ln t\to +\infty$ as any power goes to infinity faster than $\ln$, and the second term $\frac{e^{t-\sqrt{t}}}{\ln t}$ goes to $+\infty$ as $t$ wins w.r.t. $\sqrt{t}$, and the exponential goes to infinity faster than $\ln$. Therefore the requested limit exists and is equal to $-\infty$.