Use Riemann integral to evaluate the limit $\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^n \sqrt{k}}{\sum_{k=1}^n \sqrt{n+k}}$

Question

Use Riemann integral to evaluate the limit $$\lim_{n\to\infty}\frac{\sum_{k=1}^n \sqrt k}{\sum_{k=1}^n\sqrt{n+k}}$$

Answer 1

Hint. We have that $$\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^n \sqrt{k}}{\sum_{k=1}^n \sqrt{n+k}}= \lim_{n \rightarrow \infty}\frac{\displaystyle\frac{1}{n}\sum_{k=1}^n \sqrt{\frac{k}{n}}}{\displaystyle\frac{1}{n}\sum_{k=1}^n \sqrt{1+\frac{k}{n}}} = \frac{\displaystyle\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{k=1}^n \sqrt{\frac{k}{n}}}{\displaystyle\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{k=1}^n \sqrt{1+\frac{k}{n}}}=\frac{\int_0^1 f(x)\,dx}{\int_0^1 g(x)\, dx}.$$ What are the integrand functions $f$ and $g$? What is the final result?