Question

$f\left(x\right)=\{\begin{array}{cc}\frac{\cos 3x-cos4x}{x^2}& forx\ne 0\\ \frac{7}{2}& forx=0\end{array}$
at $x=0$ is

• A. continuous
• B. discontinous
• C. left continous
• D. right continous

Answer 1

The correct option is A continuous

LHL$={lim}_{x\to 0^{-}}\frac{\cos 3x-\cos 4x}{x^2}$

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

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

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

$=\frac{7}{2}$

RHL$={lim}_{x\to 0^{+}}\frac{\cos 3x-\cos 4x}{x^2}$

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

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

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

$=\frac{7}{2}$

$\therefore LHL=RHL$ and $f\left(x\right)=\frac{7}{2}$ at $x=0$

Hence $f\left(x\right)$ is continuous at $x=0$