# Limit of <msqrt> 1 &#x2212;<!-- − --> cos &#x2061;<!-- ⁡ -->

Limit of $\frac{\sqrt{1-\mathrm{cos}x}}{x}$ using l'Hôpital
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Zackery Harvey
HINT
I would say: $\phantom{\rule{1em}{0ex}}\underset{x\to 0}{lim}\frac{\sqrt{1-\mathrm{cos}\left(x\right)}}{x}={\left(\underset{x\to 0}{lim}\frac{1-\mathrm{cos}\left(x\right)}{{x}^{2}}\right)}^{1/2}=\cdots$

ddcon4r
$1-\mathrm{cos}x=2{\mathrm{sin}}^{2}\frac{x}{2}$ , so:
$\underset{x\to 0}{lim}\frac{\sqrt{1-\mathrm{cos}x}}{x}=\underset{x\to 0}{lim}\frac{\sqrt{2}|\mathrm{sin}\frac{x}{2}|}{x}$
$=\underset{x\to 0}{lim}\frac{\frac{1}{\sqrt{2}}|\mathrm{sin}\frac{x}{2}|}{\frac{x}{2}}$
$=\frac{1}{\sqrt{2}}$