Can I conclude that $\lim_{x\to0^+}\frac{x^2}{e^{-\frac{1}{x^2}}\cos(\frac{1}{x^2})^2}$ is infinite or it doesn't exists?

Question

$$\lim_{x\to0^+}\frac{x^2}{e^{-\frac{1}{x^2}}\cos(\frac{1}{x^2})^2}$$

My intuition is that the denominator goes to 0 faster and everything is non-negative, so the limit is positive infinity.

I cant think of elementary proof, l'Hopital doesn't help here, and I'm not sure if and how to use taylor.

WolframAlpha says it's complex infinity, but I can't understand why it doesn't just real positive infinity.

I've tried to use wolfram language to use the assumption that x is real, but couldn't get any result.

Answer 1

The limit$$\lim_{x\to0^+}\frac{x^2}{\exp\left(-\frac1{x^2}\right)}$$is indeed $+\infty$. Since $\dfrac1{\cos^2\left(\frac1{x^2}\right)}\geqslant1$ for each $x$ (when it is defined), you are right: the limit is $+\infty$.

Answer 2

The function is not finite valued. If you allow the value $\infty$ then the limit is $\infty$. Use L'Hôpital's Rule twice to show that $\frac {e^{y}} {y^{2}} \to \infty$ as $y \to \infty$. Replace $y$ by $\frac 1 {x^{2}}$ and use the fact that $|\cos (t)| \leq 1$.