# Limit <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAtom-ORD

Limit $\underset{x\to \mathrm{\infty }}{lim}\left(\mathrm{sin}\sqrt{1+x}-\mathrm{sin}\sqrt{x}\right)$
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Sawyer Day
$\mathrm{sin}\sqrt{x+1}-\mathrm{sin}\sqrt{x}=2\mathrm{cos}\frac{\sqrt{x+1}+\sqrt{x}}{2}\mathrm{sin}\frac{\sqrt{x+1}-\sqrt{x}}{2}$
Now
$\underset{x\to \mathrm{\infty }}{lim}\frac{\sqrt{x+1}-\sqrt{x}}{2}=0$
and
$\mathrm{cos}\frac{\sqrt{x+1}+\sqrt{x}}{2}$
is bounded.