Question

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

• A. $4$
• B. $2$
• C. $\frac{1}{2}$
• D. $1$

Answer 1

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

Now,

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

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

$=\frac{1}{2}\underset{x\to 0}{lim}\frac{{\sin }^2\frac{x}{2}}{\frac{x^2}{4}}$

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

$=\frac{1}{2}.1$

$=\frac{1}{2}$.