Question Number 75933 by Rio Michael last updated on 21/Dec/19 $${solve}\:{the}\:{inequality} \\ $$$${a}.\:\:{ln}\left(\mathrm{2}{x}−{e}\right)\:>\mathrm{1} \\ $$$${b}.\:\left({lnx}\right)^{\mathrm{2}} −{lnx}−\mathrm{6}<\mathrm{0} \\ $$$${c}.\:\mid{x}\mid\:+\:\mid{x}+\mathrm{2}\mid\:\geqslant\:\mathrm{2} \\ $$$${d}.\:\mid\mathrm{2}{x}−\mathrm{5}\mid\:+\:\mid{x}\:+\mathrm{2}\mid\:>\:\mathrm{7} \\ $$ Commented by mathmax…
Question Number 75930 by Rio Michael last updated on 21/Dec/19 $${consider}\:{the}\:{function} \\ $$$$\:{f}\left({x}\right)\:=\:\begin{cases}{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} ,\:{if}\:{x}\:{is}\:{rational}}\\{\mathrm{1}\:+\:{x}^{\mathrm{4}} ,\:{if}\:{x}\:{is}\:{irrational}}\end{cases} \\ $$$${Use}\:{the}\:{sandwich}\left({pinchin}\right)\:{theorem}\:{to} \\ $$$${prove}\:{that}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{1}. \\ $$ Terms of Service…
Question Number 75931 by Rio Michael last updated on 21/Dec/19 $${Evaluate} \\ $$$${a}.\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\sqrt[{\mathrm{3}}]{\frac{{x}^{\mathrm{2}} +\mathrm{3}}{\mathrm{27}{x}^{\mathrm{2}} −\mathrm{1}}} \\ $$$${b}.\:\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\frac{{x}−\mathrm{2}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$$${c}.\underset{{x}\rightarrow\infty} {\:\mathrm{lim}}\frac{{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{2}{x}−\mathrm{3}} \\…
Question Number 10394 by Ingjberry last updated on 07/Feb/17 $$\int\mathrm{x}×\sqrt{\mathrm{x}\:\mathrm{dx}=} \\ $$ Answered by nume1114 last updated on 07/Feb/17 $${is}\:{it}\:\int{x}\sqrt{{x}}{dx}\:?? \\ $$$$\:\:\:\:\int{x}\sqrt{{x}}{dx} \\ $$$$=\int{x}^{\mathrm{3}/\mathrm{2}} \\…
Question Number 75929 by Rio Michael last updated on 21/Dec/19 $${Evaluate} \\ $$$$\underset{{x}\rightarrow−\infty} {\:\mathrm{lim}}\:\left[\sqrt{\mathrm{1}−{xe}^{{x}} }\:\right] \\ $$ Commented by kaivan.ahmadi last updated on 21/Dec/19 $${lim}_{{x}\rightarrow−\infty}…
Question Number 10392 by ketto last updated on 06/Feb/17 $${find}\:{the}\:{direction}\:{cosines}\: \\ $$$${and}\:{its}\:{angles}\:{on} \\ $$$$\mathrm{2}{i}\:−\:\mathrm{3}{j}\: \\ $$ Answered by mrW1 last updated on 07/Feb/17 $$\mathrm{cos}\:\alpha=\frac{\mathrm{2}}{\:\sqrt{\mathrm{2}^{\mathrm{2}} +\left(−\mathrm{3}\right)^{\mathrm{2}}…
Question Number 10391 by 314159 last updated on 06/Feb/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{through}\:\left(\mathrm{2},\mathrm{3}\right)\: \\ $$$$\left(\mathrm{i}\right)\mathrm{parallel}\:\mathrm{to} \\ $$$$\left(\mathrm{ii}\right)\mathrm{perpendicular}\:\mathrm{to}\:\mathrm{2x}−\mathrm{3y}+\mathrm{6}=\mathrm{0} \\ $$ Answered by mrW1 last updated on 06/Feb/17…
Question Number 141463 by bramlexs22 last updated on 19/May/21 $$\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} \left(\:\sqrt[{\mathrm{7}\:\:}]{\frac{{x}^{\mathrm{3}} +{x}}{{x}^{\mathrm{3}} +\mathrm{1}}}\:−\mathrm{cos}\:\frac{\mathrm{1}}{{x}}\right)? \\ $$ Commented by jcarlos last updated on 19/May/21 $$\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}}…
Question Number 141457 by gajmer last updated on 19/May/21 Answered by mathmax by abdo last updated on 19/May/21 $$\mathrm{f}\left(\mathrm{x}\right)=\frac{\sqrt{\mathrm{2x}^{\mathrm{2}} +\mathrm{4}}−\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{5}}}{\mathrm{x}−\mathrm{1}}\:\Rightarrow \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mid\mathrm{x}\mid\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\frac{\mathrm{2}}{\mathrm{x}^{\mathrm{2}} }}−\mid\mathrm{x}\mid\sqrt{\mathrm{1}+\frac{\mathrm{5}}{\mathrm{x}^{\mathrm{2}} }}}{\mathrm{x}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}\right)}\:\Rightarrow…
Question Number 10387 by FilupSmith last updated on 06/Feb/17 $$\mathrm{6}\:\mathrm{people}\:{a},\:{b},\:{c},\:{d},\:{e},\:\mathrm{and}\:{f}\:\mathrm{stand}\:\mathrm{in}\:\mathrm{a}\:\mathrm{line}. \\ $$$$\: \\ $$$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{they}\:\mathrm{can}\:\mathrm{stand}\:\mathrm{arranged} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{6}! \\ $$$$\: \\ $$$$\mathrm{If}\:\mathrm{two}\:\mathrm{people}\:\mathrm{have}\:\mathrm{to}\:\mathrm{stand}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}, \\ $$$$\mathrm{but}\:\mathrm{everyone}\:\mathrm{else}\:\mathrm{do}\:\mathrm{not}\:\mathrm{matter},\:\mathrm{how}\:\mathrm{many}\:\mathrm{combinations} \\ $$$$\mathrm{combinations}\:\mathrm{are}\:\mathrm{there}? \\…