Result of the limit: $\lim_{x \to \infty} \sqrt{x + \sin(x)} - \sqrt{x}$? From my calculations, the limit of 
$\lim_{x \to \infty} \sqrt{x + \sin(x)} - \sqrt{x}$
Is undefined due to $sin(x)$ being a periodic function, but someone told me it should be zero. 
I was just wondering if someone could please confirm what the limit of this function is? Thanks
Corey :) 

or use the binomial theorem
$$
\sqrt{x+\sin x}-\sqrt{x} = \sqrt{x} \left(\sqrt{1 + \frac{\sin x}
{x}}-1\right) \\
= \sqrt{x}\left( \frac{\sin x}x + O(\frac1{x^2}) \right) \to 0
$$
