Question

The function $f\left(x\right)=\left\{\begin{array}{lc}\frac{\sin x}{x}+\cos x,& \mathrm{if}x\ne 0\\ k,& \mathrm{if}x=0\end{array}\right.$ is continuous at x = 0, then the value of k is
(a) 3
(b) 2
(c) 1
(d) 1.5

Answer 1

It is given that, the function $f\left(x\right)=\left\{\begin{array}{lc}\frac{\sin x}{x}+\cos x,& \mathrm{if}x\ne 0\\ k,& \mathrm{if}x=0\end{array}\right.$ is continuous at x = 0.

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

$\Rightarrow k=\underset{x\to 0}{\lim }\left(\frac{\sin x}{x}+\cos x\right)$

$\Rightarrow k=\underset{x\to 0}{\lim }\frac{\sin x}{x}+\underset{x\to 0}{\lim }\cos x$

$\Rightarrow k=1+1=2$

Thus, the value of k is 2.

Hence, the correct answer is option (b).