Question

Evaluate $\underset{x\to 0}{lim}\frac{\sqrt{1-\cos x}}{x}$

• A. is equal to $\frac{1}{\sqrt{2}}$
• B. is equal to $\sqrt{2}$
• C. is equal to $0$
• D. does not exist

Answer 1

The correct option is A is equal to $\frac{1}{\sqrt{2}}$

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

$=\underset{x\to 0}{lim}\frac{\sqrt{1-1+2{\sin }^2\frac{x}{2}}}{2\frac{x}{2}}$......................[$cos2x=1-2si{n}^{2}x$]

$=\underset{x\to 0}{lim}\frac{\sqrt{2}\sin \frac{x}{2}}{2\frac{x}{2}}=\frac{1}{\sqrt{2}}$..................[since, $\underset{x\to 0}{lim}\frac{sinx}{x}=1$]