lim_x→ 0 √(1 cos2x2)x lim_x→ 0√(2sin^2x2)x #160; as cos^2x=1 2sin^2x ⇒lim_x→ 0sin xx=1 ...

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Substitution Method to Remove Indeterminate Form

limx→ 0√1-cos...

Question

$lim x \to 0 \frac{\sqrt{\frac{1 - cos 2 x}{2}}}{x}$ =

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Solution

${lim}_{x \to 0}$ $\frac{\sqrt{\frac{1 - cos 2 x}{2}}}{x}$
${lim}_{x \to 0} \frac{\sqrt{\frac{2 {sin}^{2} x}{2}}}{x}$ as ${cos}^{2} x = 1 - 2 {sin}^{2} x$
$\Rightarrow {lim}_{x \to 0} \frac{sin x}{x} = 1$

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