I have trouble finding the value of the following limit: $$\lim_{n \to \infty} \sqrt{n} \sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)$$
For now I have rewritten the term into: $$ \lim_{n \to \infty} \dfrac{\sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)}{\large \frac{1}{\sqrt{n}}}$$ Now I have a limit of type $\large \frac{0}{0}$ so I think I could use L'Hopital's rule. But I would like to know if there is a way you can solve this without using L'Hopital's rule.
Thanks for your answers.