Question

If f : R → R defined by $f\left(x\right)=\left\{\begin{array}{cc}\frac{\cos 3x-\cos x}{x^2},& x\ne 0\\ \lambda ,& x=0\end{array}\right.$ is continuous at x = 0, then λ = _____________.

Answer 1

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

$\therefore f\left(0\right)=\underset{x\to 0}{\lim }f\left(x\right)$

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

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

$\Rightarrow \lambda =\underset{x\to 0}{\lim }\frac{-2\sin 2x\sin x}{x^2}$

$\Rightarrow \lambda =-2\times \underset{x\to 0}{\lim }\frac{\sin 2x}{x}\times \underset{x\to 0}{\lim }\frac{\sin x}{x}$

$\Rightarrow \lambda =-2\times 2\underset{x\to 0}{\lim }\frac{\sin 2x}{2x}\times \underset{x\to 0}{\lim }\frac{\sin x}{x}$

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

$\Rightarrow \lambda =-4$

Thus, the value of λ is −4.


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