Find the limit $\lim\limits_{n \to \infty}\sum_{k=1}^n \frac{\sqrt{4n+5k}}{n \sqrt{n}} $

Question

I got stuck on the following limit:

$\lim\limits_{n \to \infty}5\sum_{k=1}^n \frac{\sqrt{4n+5k}}{n \sqrt{n}} $

The expected answer is $\frac{38}{15}.$ Any hint is appreciated. Thanks

EDIT: The task is from this polish textbook (http://lucc.pl/inf/analiza_1/gewert_skoczylas__analiza_matematyczna_1__definicje_twierdzenia_wzory.pdf). It's on page 166 of the book (ex. 8.4.3 h). I want to compute this using the above fact. It's a book for first year students.

Answer 1

Hint (Riemman Sum-Integral) :

$$\lim_{n \to \infty} \sum_{k=1}^n \frac{\sqrt{4n+5k}}{n\sqrt{n}} = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \sqrt{4 + 5\frac{k}{n}} = \int_0^1 \sqrt{4 + 5x} \;\mathrm{d}x$$