Limit of a Sum

Calculus Level 4

$\lim_{x\to\infty} \dfrac{1}{x} \left(\dfrac{1}{3+\dfrac{1}{x}}+\dfrac{1}{3+\dfrac{2}{x}}+\dfrac{1}{3+\dfrac{3}{x}}+\cdots+\dfrac{1}{3+\dfrac{x}{x}}\right) = \, ?$

Give your answer rounded to 3 decimal places.

If you come to the conclusion that the limit diverges, enter 1.772.