Evaluate a limit

Question

I've been stuck on this limit $$\lim_{x\to 0} \frac{e^{-1/x²}}{x²}$$ i've tried evaluating from both sides, and got $\lim_{x\to 0^-}\frac{1}{e^{1/x²}}\times \frac{1}{x²}$ which actually gives 0 times something that tends to infinity. Same thing happens with the right hand limit, not sure from where to go from here, i don't see any possible cancellations or anything to get rid of 1/x².

Answer 1

Without L'Hospital or Taylor:

$e^x=\underset{n\to \infty}\lim\left(1+\frac xn\right)^n$. It's easy to show that the sequence is increasing so $e^x\ge \left(1+\frac xn\right)^n\ge 1+x.$ So, $1+x\le e^x$ and so $0\le e^{-x}\le \frac{1}{1+x}.$ Thus, $0\le e^{-x^2}\le \frac{1}{1+x^2}\Rightarrow 0\le x^2e^{-x^2}\le \frac{x^2}{1+x^2}$ which implies that $x^2e^{-x^2}\to 0$ as $x\to 0.$ But this is exactly the limit you want, once you make the substitution $t=1/x.$

Answer 2

$$L=\lim_{x\to 0} \frac{e^{-1/x^2}}{x^2}$$ Let $x=1/t$, then $$L=\lim_{t\to \infty} t^2e^{-t^2}=\lim_{t \to \infty} \frac{t^2}{e^{t^2}}\to \frac{0}{0}$$ Use L-Hospital, then $$L=\lim_{t \to \infty} \frac{2t}{2t e^{t^2}}=\lim_{t\to \infty} e^{-t^2}=0$$