Question

If $f\left(x\right)=\left\{\frac{1-\cos x}{\begin{array}{c}x\\ k\end{array}}\begin{array}{c}\\ x\ne 0\\ \\ x=0\end{array}\right\}$ is continuous at $x=0$, then the value of $k$ is:

• A. $0$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $-\frac{1}{2}$

Answer 1

The correct option is A $0$

For $f$ to be continuous $\text{f}\left(0\right)=\underset{x\to 0}{lim}\text{f}\left(x\right)$

$\Rightarrow k=\underset{x\to 0}{lim}\frac{1-\cos x}{x}$

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