Question Number 77533 by vishalbhardwaj last updated on 07/Jan/20
$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\:\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:\mathrm{where}\:\mathrm{x}\neq\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{function}\:\mathrm{at}\:\mathrm{x}=\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{given}\:\mathrm{that}\:\mathrm{function}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{at}\:\mathrm{x}=\mathrm{0}\:? \\ $$
Answered by MJS last updated on 07/Jan/20
![lim_(x→0) xsin (1/x) = [let x=(1/t)] =lim_(t→±∞) ((sin t)/t) =0 ⇒ f(0)=0](Q77534.png)
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}\mathrm{sin}\:\frac{\mathrm{1}}{{x}}\:= \\ $$$$\:\:\:\:\:\left[\mathrm{let}\:{x}=\frac{\mathrm{1}}{{t}}\right] \\ $$$$=\underset{{t}\rightarrow\pm\infty} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{t}}{{t}}\:=\mathrm{0} \\ $$$$\Rightarrow\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$
Commented by vishalbhardwaj last updated on 07/Jan/20
thanks a lot sir