Question

Prove that
$f\left(x\right)=\{\begin{array}{c}\frac{1-\cos x}{x^2},whenx\ne 3\\ 1,whenx=3\end{array}$ is discountinuous at $x=0$.

Answer 1

$={lim}_{x\to 3}\frac{1-\cos x}{x^2}$

$={lim}_{x\to 3}\frac{2{\sin }^2\frac{x}{2}}{{\left(\frac{x}{2}\right)}^2\times 4}$

$=2\times \frac{1}{4}=\frac{1}{2}$

$\therefore f\left(3\right)=1\ne {lim}_{x\to 3}\frac{1-\cos x}{x^2}$

Hence the function is not continuous at $x=3$