Existence or regularization of the limit: $\lim_{\beta\to\infty} \frac{\beta^2}{\sin(\beta^2)}$

Question

I came across this limit and I really do think it is indeterminate or it doesn't exist:

$$\lim_{\beta\to \infty} \frac{\beta^2}{\sin(\beta^2)}$$

The Sine function is bounded, and besides that, using $x = 1/\beta^2$ I get

$$\lim_{x\to 0} \frac{1}{x\sin\left(\frac{1}{x}\right)}$$

Which seems not to exist.

Is there some rigorous proof about this?

Or is there some "definition or regularization" for that limit?

Thank you!

Answer 1

The point is, if we let $\beta_{n}=\sqrt{2n\pi}$, then $\dfrac{\beta_{n}^{2}}{\sin(\beta_{n}^{2})}=\dfrac{2n\pi}{0}$, $n=1,2,...$ which is not defined infinitely many terms.

Meanwhile, $\beta_{n}=\sqrt{2n\pi+\dfrac{\pi}{6}}$, then $\dfrac{\beta_{n}^{2}}{\sin(\beta_{n}^{2})}=4n\pi+\dfrac{\pi}{3}\rightarrow\infty$.