Question

If $f:R\to R$ is defined by $f\left(x\right)=\left\{\begin{array}{cc}\frac{\cos 3x-\cos x}{x^2}& ,forx\ne 0\\ \lambda ,& ,forx=0\end{array}\right\}$ and if $f$ is continuous at $x=0$, then $\lambda$ is equal to

• A. $-2$
• B. $-4$
• C. $-6$
• D. $-8$

Answer 1

Answer: (B)

$-4$

Given that,
$f\left(x\right)=\left\{\begin{array}{cc}\frac{\cos 3x-\cos x}{x^2}& forx\ne 0\\ \lambda & forx=0\end{array}\right\}$
Now, LHL $=\underset{x\to 0^{-}}{lim}f\left(x\right)$
$=\underset{x\to 0^{-}}{lim}\frac{\cos 3x-\cos x}{x^2}$
$=\underset{h\to 0}{lim}\frac{\cos 3(0-h)-\cos 0-h}{0-h^2}$
$=\underset{h\to 0}{lim}\frac{\cos 3h-\cos h}{h^2}$
$=\underset{h\to 0}{lim}\frac{-3\sin 3h+\sin h}{2h}$ (using L'Hospitals' rule)
$=\frac{-9+1}{2}=-4$
Since, $f\left(x\right)$ is continuous at $x=0$
$\therefore \underset{x\to 0^{-}}{lim}f\left(x\right)=f\left(0\right)$
$\Rightarrow -4=\lambda$

$\Rightarrow \lambda =-4$