calculus - $limlimits_{xtoinfty } frac{ln x}{sqrt{x},{sin{x}}}$

What a weird function.

I tried to find out:

$$\lim_{x\to\infty } \frac{\ln x}{\sqrt{x}\,{\sin{x}}}$$

So, I can't use L'Hopital 'cause there's no actual limit in the denominator. It doesn't exist.

Then, I tried to use Heine's theorem and chose two sequences, but yet I got the same limit.

I believe it does not converge. How can I prove it?

Thanks

Answer

You can consider a sequence $x_n = \pi n - 2^{-n}$. On this sequence your function will become unbounded while this sequence goes to infinity. Hence the limit does not exist.