Question

The value of $\underset{x\to 0}{lim}\frac{1-\cos x}{x^2}$ equals

• A. 2
• B. 1
• C. $\frac{1}{2}$
• D. 0

Answer 1

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

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

Now we are going to use L'hospital rule,

$\underset{x\to a}{lim}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{lim}\frac{f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)}$

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

$\Rightarrow$ $\frac{1}{2}\underset{x\to 0}{lim}\frac{\sin x}{2x}$

$\Rightarrow$ $\frac{1}{2}\times \left(1\right)$

$\Rightarrow$ $\frac{1}{2}$

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