Question

$\underset{x\to 0}{lim}\frac{1-\cos x}{x\sqrt{x^2}}$ equal to

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $Zero$
• D. $Doesnotexist$

Answer 1

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

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

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

$=\underset{x\to 0}{lim}\frac{2{\sin }^2\frac{x}{2}}{x^2}$

$=\underset{x\to 0}{lim}\frac{2{\sin }^2\frac{x}{2}}{{\left(\frac{x}{2}\right)}^2}\times \frac{1}{4}$

$=2\times 1\times \frac{1}{4}=\frac{1}{2}$