$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$

Question

I was trying to do this limit $$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$$ and I cant, I will be pleased if someone give me a hint to do it.

Answer 1

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

Answer 2

$$\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}$$ hence the limit is $\frac{1}{2}$.