Question

Discuss the countinuity of the following function at $x=0$
$f\left(x\right)=\{\begin{array}{c}\frac{1-\cos x}{x^2},\\ \frac{1}{2},\end{array}\begin{array}{c}x\ne 0\\ x=0\end{array}$

Answer 1

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

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

As $\underset{\theta \to 0}{lim}\frac{\sin \theta }{\theta }=1$

$\Rightarrow L=\underset{x\to 0}{lim}\frac{1}{2}\times 1=\frac{1}{2}$

$\Rightarrow \underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)=f$ is continuous at $x=0$.