Evaluate the limit $\lim_{n\to\infty} n(\sqrt{n}-\sqrt{n+1})$ Evaluate the limit $\lim_{n\to\infty} n(\sqrt{n}-\sqrt{n+1})$
I am trying to evaluate this limit. First I multiplied by the conjugate to obtain: $a_n=\dfrac{-n}{\sqrt{n}+\sqrt{n+1}}$
I was able to show the limit by taking $f(x)=a_x$ and then applying l'Hôpital's rule. I was primarily wondering if there is a certain trick that can be used to find the limit without resorting to this.

 \begin{align} \lim_{n\to\infty} n(\sqrt{n}-\sqrt{n+1})=-\infty \end{align}

Thanks.

Start from
$${-n\over \sqrt{n}+\sqrt{n+1}}={-\sqrt{n}\over 1+\sqrt{1+{1\over n}}}={N\over D}$$
We have $D\to 1$ and $N\to -\infty$
