Question

The value of k which makes
$f\left(x\right)=\left\{\begin{array}{ll}\sin \frac{1}{x},& x\ne 0\\ k,& x=0\end{array}\right.$continuous at x = 0, is
(a) 8
(b) 1
(c) −1
(d) none of these

Answer 1

(d) none of these

If $f\left(x\right)$ is continuous at $x=0$, then
$$\begin{gathered} \underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right) \\ \Rightarrow \underset{x\to 0}{\lim }\left(\sin \frac{1}{x}\right)=k \\ \mathrm{But}\underset{x\to 0}{\lim }\left(\sin \frac{1}{x}\right)\mathrm{does}\mathrm{not}\mathrm{exist}.\mathrm{Thus},\mathrm{there}\mathrm{does}\mathrm{not}\mathrm{exist}\mathrm{any}k\mathrm{that}\mathrm{makes}f\left(x\right)\mathrm{a}\mathrm{continuous}\mathrm{function}. \end{gathered}$$