Question

Show that $f\left(x\right)=\frac{\cos 3x-\cos 4x}{x\sin 2x}$ for $x\ne 0,f\left(0\right)=\frac{7}{4}$ is continuous at $x=0$

Answer 1

${lim}_{x\to 0}f\left(x\right)={lim}_{x\to 0}\frac{\cos 3x-\cos 4x}{x\sin 2x}$

$={lim}_{x\to 0}\frac{2\sin \left(\frac{7x}{2}\right)\sin \left(\frac{x}{2}\right)}{x\sin 2x}$

$=2{lim}_{x\to 0}\frac{\sin \left(\frac{7x}{2}\right)}{x}\frac{\sin \left(\frac{x}{2}\right)}{x}\times \frac{x}{\sin 2x}$

$=2\times \frac{7}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{7}{4}=f\left(0\right)$

So, $f\left(x\right)$ is continuous at $x=0$.