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 :)
$$\sqrt{x+\sin x}-\sqrt{x}=\frac{(x+\sin x)-(x)}{\sqrt{x+\sin x}+\sqrt{x}}=\frac{\frac{\sin x}{\sqrt{x}}}{\sqrt{1+\frac{\sin x}{x}}+1}\to \frac{0}{\sqrt{1+0}+1}= 0$$
I think the limit is $0$, since $sin(x)$ is bounded: At best, it is $1$ or $-1$, but if $x \rightarrow \infty$, a bounded number will look like a const; it won't influence the limit, so basically you have $\sqrt{x}- \sqrt{x}= 0$