If f : R → R defined by

Question:

If $f: R \rightarrow R$ defined by $f(x)=\left\{\begin{array}{cl}\frac{\cos 3 x-\cos x}{x^{2}}, & x \neq 0 \\ \lambda, & x=0\end{array}\right.$ is continuous at $x=0$, then $\lambda=$___________

Solution:

The function $f(x)=\left\{\begin{array}{cl}\frac{\cos 3 x-\cos x}{x^{2}}, & x \neq 0 \\ \lambda, & x=0\end{array}\right.$ is continuous at $x=0$

$\therefore f(0)=\lim _{x \rightarrow 0} f(x)$

$\Rightarrow \lambda=\lim _{x \rightarrow 0} \frac{\cos 3 x-\cos x}{x^{2}}$

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

$\Rightarrow \lambda=\lim _{x \rightarrow 0} \frac{-2 \sin 2 x \sin x}{x^{2}}$

$\Rightarrow \lambda=-2 \times \lim _{x \rightarrow 0} \frac{\sin 2 x}{x} \times \lim _{x \rightarrow 0} \frac{\sin x}{x}$

$\Rightarrow \lambda=-2 \times 2 \lim _{x \rightarrow 0} \frac{\sin 2 x}{2 x} \times \lim _{x \rightarrow 0} \frac{\sin x}{x}$

$\Rightarrow \lambda=-2 \times(2 \times 1) \times 1$    $\left(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right)$

$\Rightarrow \lambda=-4$

If $f: R \rightarrow R$ defined by $f(x)=\left\{\begin{array}{cl}\frac{\cos 3 x-\cos x}{x^{2}}, & x \neq 0 \\ \lambda, & x=0\end{array}\right.$ is continuous at $x=0$, then $\lambda=$ ____−4____.