Question Number 5286 by FilupSmith last updated on 05/May/16
$$\mathrm{Can}\:\mathrm{someone}\:\mathrm{please}\:\mathrm{explain}\:\mathrm{why}: \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left({x}−{x}^{\mathrm{2}} \right)\:=\:−\infty \\ $$
Commented by FilupSmith last updated on 05/May/16
$$\mathrm{i}\:\mathrm{normally}\:\mathrm{undwrstand}\:\mathrm{limits}\:\mathrm{but} \\ $$$$\mathrm{i}\:\mathrm{dont}\:\mathrm{remember}\:\mathrm{how}\:\mathrm{to}\:\mathrm{do}\:\mathrm{this} \\ $$
Answered by 123456 last updated on 05/May/16
$$\mathrm{write}\:\mathrm{it}\:\mathrm{as} \\ $$$$−{x}^{\mathrm{2}} +{x}={x}^{\mathrm{2}} \left(−\mathrm{1}+\frac{\mathrm{1}}{{x}}\right) \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}−{x}^{\mathrm{2}} \\ $$$$=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} \left(−\mathrm{1}+\frac{\mathrm{1}}{{x}}\right) \\ $$$$\frac{\mathrm{1}}{{x}}\rightarrow\mathrm{0}\:\mathrm{when}\:{x}\rightarrow\infty\:\mathrm{so} \\ $$$$=\underset{{x}\rightarrow+\infty} {\mathrm{lim}}−{x}^{\mathrm{2}} =−\infty \\ $$
Commented by FilupSmith last updated on 05/May/16
$${Thanks}!!! \\ $$
Answered by enigmeyou last updated on 18/Jun/16
$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\left({x}−{x}^{\mathrm{2}} \right)=\underset{{x}\rightarrow−\infty} {\mathrm{lim}}{x}\left(\mathrm{1}−{x}\right)=−\infty×\left(+\infty\right)=−\infty \\ $$$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left({x}−{x}^{\mathrm{2}} \right)=\underset{{x}\rightarrow+\infty} {\mathrm{lim}}{x}\left(\mathrm{1}−{x}\right)=+\infty×\left(−\infty\right)=−\infty \\ $$$$ \\ $$