I have been trying to solve the following question from my Mathematical Analysis course in university:
$$\lim \limits_{x \to +\infty}\sqrt{x}(\sqrt{x+1}-\sqrt{x-1})$$
I am aware that the answer is 1, but I am not entirely sure why.
Substituting positive infinity into the equation, if I am not terribly mistaken, gives $\infty-\infty$, which is indeterminate. Anyhow, I am not aware of any properties or theorems that could get rid of this issue, and Landau symbols don't really seem to help. I have tried as well to rewrite the problem as $$\lim \limits_{x \to +\infty}\frac{\sqrt{x+1}-\sqrt{x-1}}{\frac{1}{\sqrt{x}}}$$
to use L'Hopital's theorem, which does not get rid of the indeterminate form. Help would be appreciated. Thanks in advance!