Question

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

• A. $2$
• B. $3$
• C. $\frac{1}{2}$
• D. $\frac{1}{3}$

Answer 1

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

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

$=\underset{x\to 0}{lim}\frac{2{\sin }^2\frac{x}{2}}{4{\left(\frac{x}{2}\right)}^2}=\frac{1}{2}\underset{x\to 0}{lim}{\left(\frac{\sin \frac{x}{2}}{\frac{x}{2}}\right)}^2=\frac{1}{2}\cdot 1=\frac{1}{2}$ (using basic limit formula )