Relations between $\sqrt x$ and $\sqrt{x+n}$

Question

Is there any relation between $\sqrt x$ and $\sqrt{x+n}$? I am interested in the fractional part mostly.

n and x are both positive integers, n is much greater than x.

Answer 1

If you integrate $\int_x^{x+n} 1(/\sqrt{t})dt$ you get $2(\sqrt{x+n} - \sqrt{x})$ which for large $x$ compared to $n$ is a good approximation to the integral and so the difference $\sqrt{x+n} - \sqrt{x}$ is comparable to $(n/2)/\sqrt{x}$.