Limit $\underset{x\to \mathrm{\infty}}{lim}(\mathrm{sin}\sqrt{1+x}-\mathrm{sin}\sqrt{x})$

Damon Stokes
2022-06-26
Answered

Limit $\underset{x\to \mathrm{\infty}}{lim}(\mathrm{sin}\sqrt{1+x}-\mathrm{sin}\sqrt{x})$

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Sawyer Day

Answered 2022-06-27
Author has **30** answers

$\mathrm{sin}\sqrt{x+1}-\mathrm{sin}\sqrt{x}=2\mathrm{cos}\frac{\sqrt{x+1}+\sqrt{x}}{2}\mathrm{sin}\frac{\sqrt{x+1}-\sqrt{x}}{2}$

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.

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.

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