Limit $\lim_{x \rightarrow \infty} (\sin {\sqrt{1+x}} - \sin {\sqrt{x}})$ I need to calculate limit:
$\lim_{x \rightarrow \infty} (\sin {\sqrt{1+x}} - \sin {\sqrt{x}})$
I was thinking of using the formula for $\sin \alpha - \sin \beta$, but what can be the next step?

$$
\sin\sqrt{x+1}-\sin\sqrt{x}=
2\cos\frac{\sqrt{x+1}+\sqrt{x}}{2}
\sin\frac{\sqrt{x+1}-\sqrt{x}}{2}
$$
Now
$$
\lim_{x\to\infty}\frac{\sqrt{x+1}-\sqrt{x}}{2}=0
$$
and
$$
\cos\frac{\sqrt{x+1}+\sqrt{x}}{2}
$$
is bounded.
