Show that $f\left(x\right)=\left\{\begin{array}{cll}\frac{\cos ax-\cos bx}{x^2}& \text{ if }& x\ne 0\\ \frac{1}{2}\left(b^2-a^2\right)& \text{ if }& x=0\end{array}\right.$where a and b are real constants, is continuous at $x=0\text{.}$

Given that  $f\left(x\right)=\left\{\begin{array}{cl}\frac{\cos ax-\cos bx}{x^2}& \text{ if }x\ne 0\\ \frac{1}{2}\left(b^2-a^2\right)& \text{ if }x=0\end{array}\right.$

$\underset{x\to 0}{lt}f\left(x\right)={\mathrm{lt}}_{x\to 0}\frac{\cos ax-\cos bx}{x^2}$

$=\underset{x\to 0}{lt}\frac{2\sin \left(\frac{bx+ax}{2}\right)\sin \left(\frac{bx-ax}{2}\right)}{x^2}$

$=\underset{x\to 0}{lt}\frac{2\sin \left(\frac{b+a}{2}\right)x}{x}\frac{\sin \left(\frac{b-a}{2}\right)x}{x}$

$=\frac{2(b+a)}{2}\frac{(b-a)}{2}=\frac{b^2-a^2}{2}\Rightarrow \underset{x\to 0}{lt}f\left(x\right)=f\left(0\right)$

$\therefore f\left(x\right)\text{ is continuous at }x=0$