Question Number 186758 by EnterUsername last updated on 09/Feb/23
![If [t] denotes the integral part of t, then lim_(x→1) [x sin πx] (A) equals 1 (B) equals −1 (C) equals 0 (D) does not exist](https://www.tinkutara.com/question/Q186758.png)
$$\mathrm{If}\:\left[{t}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{part}\:\mathrm{of}\:{t},\:\mathrm{then}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left[{x}\:\mathrm{sin}\:\pi{x}\right] \\ $$$$\left(\mathrm{A}\right)\:\:\mathrm{equals}\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\:\mathrm{equals}\:−\mathrm{1} \\ $$$$\left(\mathrm{C}\right)\:\:\mathrm{equals}\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$
Answered by Gazella thomsonii last updated on 10/Feb/23
$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\int\:{x}\mathrm{sin}\left(\pi{x}\right)\mathrm{d}{x}=\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\pi}\right)^{\mathrm{2}} \centerdot\left(\mathrm{sin}\left(\pi{x}\right)−\pi{x}\mathrm{cos}\left(\pi{x}\right)\right. \\ $$$$\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\:\:\:\:\left(\mathrm{D}\right)\centerdot\centerdot\centerdot\centerdot\:\mathrm{Dosen}'\mathrm{t}\:\mathrm{Exist} \\ $$
Answered by mr W last updated on 10/Feb/23
![lim_(x→1^− ) [x sin πx]=0 lim_(x→1^+ ) [x sin πx]=−1 ⇒lim_(x→1) [x sin πx] doesn′t exist!](https://www.tinkutara.com/question/Q186820.png)
$$\underset{{x}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}}\left[{x}\:\mathrm{sin}\:\pi{x}\right]=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{1}^{+} } {\mathrm{lim}}\left[{x}\:\mathrm{sin}\:\pi{x}\right]=−\mathrm{1} \\ $$$$\Rightarrow\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left[{x}\:\mathrm{sin}\:\pi{x}\right]\:{doesn}'{t}\:{exist}! \\ $$
Commented by EnterUsername last updated on 10/Feb/23
Thank you, Sir.