Question

If $f\left(x\right)=\left\{\begin{array}{ll}\frac{1-\cos x}{x^2},& x\ne 0\\ k,& x=0\end{array}\right.$is continuous at x = 0, find k.

Answer 1

Given: $f\left(x\right)=\left\{\begin{array}{l}\frac{1-\cos x}{x^2},x\ne 0\\ k,x=0\end{array}\right.$
If $f\left(x\right)$ is continuous at $x=0$, then
$\underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$

$$\begin{gathered} \Rightarrow \underset{x\to 0}{\lim }\left(\frac{1-\cos x}{x^2}\right)=k \\ \Rightarrow \underset{x\to 0}{\lim }\left(\frac{2{\left[\sin \left({\displaystyle \frac{x}{2}}\right)\right]}^2}{4{\left({\displaystyle \frac{x}{2}}\right)}^2}\right)=k \\ \Rightarrow \frac{1}{2}\underset{x\to 0}{\lim }\left(\frac{{\left[\sin \left({\displaystyle \frac{x}{2}}\right)\right]}^2}{{\left({\displaystyle \frac{x}{2}}\right)}^2}\right)=k \\ \Rightarrow 1\times \frac{1}{2}=k \\ \Rightarrow k=\frac{1}{2} \end{gathered}$$