Find $\lim \limits_{n \to \infty}(\sqrt{n+1} - \sqrt{n})*n^p, p\in N$

Question

I got into some trouble with this limit which leads to an indetermination in the final part: $\lim \limits_{n \to \infty}(\sqrt{n+1} - \sqrt{n})*n^p, p\in N$

The limit in the parenthesis results $\frac{1}{\sqrt{n}}$ after amplification with $\sqrt{n+1}-\sqrt{n}$ which it's limit is $0$, but here comes the indetermination because $n$ to any power except $0$ is $+\infty$. Can someone help me please?

Answer 1

Write : $$(\sqrt{n+1}-\sqrt{n})\times n^p=\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}(\sqrt{n+1}+\sqrt{n})n^p=\frac{n^p}{\sqrt{n+1}+\sqrt{n}}=\frac{n^p}{\sqrt{n}\Big(\sqrt{1+\frac{1}{n}}+1\Big)}$$ this goes to $\infty$ as $n$ goes to infinity.