If f : R → R is defined by $f(x)=\left\{\begin{array}{cc} \frac{\cos 3 x-\cos x}{x^2}, & x \neq 0 \\ \lambda, & x=0 \end{array}\right.$ and if f is continuous at x = 0, then $\lambda$ is equal to
Answer & explanation
Correct answer: option 2
If f is continuous x = 0, then
$\lim\limits_{x \rightarrow 0} f(x)=f(0)$
$\Rightarrow \lim\limits_{x \rightarrow 0} \frac{\cos 3 x-\cos x}{x^2}=\lambda$
$\Rightarrow \lim\limits_{x \rightarrow 0} \frac{-2 \sin 2 x \sin x}{x^2}=\lambda$
$\Rightarrow -4 \lim\limits_{x \rightarrow 0}\left(\frac{\sin 2 x}{2 x}\right)\left(\frac{\sin x}{x}\right)=\lambda \Rightarrow \lambda=-4 \times 1 \times 1=-4$