Use the definition of a limit to prove the following: $\lim\limits_{x\to0}\frac{\cos x}{x^{2}}=\infty$

Question

Use the definition of a limit to prove the following: $$\lim_{x\to0}\frac{\cos x}{x^{2}}=\infty$$

I'm trying to prove that: $$\forall E >0\,\,,\exists\delta>0 \\~\\ 0<|x|<\delta\Longrightarrow\frac{\cos x}{x^{2}}>E$$ I know $-1\leq \cos x\leq1$ and $\frac{1}{x^{2}}>\frac{1}{\delta^{2}}$ ,But I can't put it together.

Answer 1

For any $\;E>0\;,\;$ take $\;\delta=\min\left\{\frac\pi3,\frac1{\sqrt{2E}}\right\}>0\;$.

Then, if $\;0<|x|<\delta\;,\;$ you have $$\cos x>\frac12\quad\text{and}\quad\frac1{x^2}>2E\;.$$ Therefore, $\;\dfrac{\cos x}{x^2}>E\;$.