Question Number 120230 by MehraGanesh last updated on 30/Oct/20 $$\:\:\:\:\int\:\frac{\mathrm{x}^{\mathrm{3}} }{\left(\mathrm{x}\:+\mathrm{2}\right)^{\mathrm{4}} }\:\mathrm{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{OR} \\ $$$$\:\:\:\int\:\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{1}\:+\:\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx} \\ $$$$ \\ $$$$\:\:\:\:\:\overset{\rightarrow} {\mathrm{a}}\:=\:\hat {\mathrm{i}}\:−\:\hat {\mathrm{j}}\:+\:\mathrm{3}\hat {\mathrm{k}}\:\:\mathrm{and}\:\overset{\rightarrow\:} {\mathrm{b}}\:=\:\mathrm{2}\hat…
Question Number 54688 by pooja24 last updated on 09/Feb/19 $${f}\left({x}\right)=\frac{\mathrm{2}\left[{x}\right]}{\mathrm{3}{x}−\left[{x}\right]}\:{examine}\:{its}\:{continuity}\:{at}\:{x}=\frac{−\mathrm{1}}{\mathrm{2}} \\ $$$${where}\:\left[{x}\right]\:{is}\:{greatest}\:{integer}\:{function} \\ $$ Commented by maxmathsup by imad last updated on 10/Feb/19 $${we}\:{have}\:{f}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\frac{\mathrm{2}\left(−\mathrm{1}\right)}{−\frac{\mathrm{3}}{\mathrm{2}}−\left(−\mathrm{1}\right)}\:=\frac{−\mathrm{2}}{−\frac{\mathrm{3}}{\mathrm{2}}+\mathrm{1}}\:=\frac{−\mathrm{2}}{−\frac{\mathrm{1}}{\mathrm{2}}}\:=\mathrm{4} \\…
Question Number 120211 by bemath last updated on 30/Oct/20 $$\:{evaluate}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}{f}\left({x}\right)\:{and}\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}{f}\left({x}\right) \\ $$$${for}\:{f}\left({x}\right)=\frac{{x}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}. \\ $$ Answered by Ar Brandon last updated on 30/Oct/20…
Question Number 120201 by bemath last updated on 30/Oct/20 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left({x}+{e}^{{x}} +{e}^{\mathrm{2}{x}} \right)}{{x}}=? \\ $$ Answered by benjo_mathlover last updated on 30/Oct/20 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left({x}+{e}^{{x}} +{e}^{\mathrm{2}{x}}…
Question Number 120068 by bramlexs22 last updated on 29/Oct/20 $$\left({i}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:{x}−\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}}{{x}^{\mathrm{4}} }\: \\ $$$$\left({ii}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{{x}} −\mathrm{1}−{x}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}}{{x}^{\mathrm{4}} } \\ $$$$\left({iii}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:{x}−{x}}{\mathrm{arc}\:\mathrm{sin}\:{x}−{x}} \\ $$…
Question Number 120067 by bramlexs22 last updated on 29/Oct/20 $$\left({i}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} \:\mathrm{sin}\:\left(\mathrm{1}/{x}\right)}{\mathrm{tan}\:{x}} \\ $$$$\left({ii}\right)\:{Without}\:{L}'{Hopital}\:{rule} \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:{x}−{x}\mathrm{sin}\:{x}}{\mathrm{2}−\mathrm{2cos}\:{x}−\mathrm{sin}\:^{\mathrm{2}} {x}} \\ $$ Answered by Olaf last…
Question Number 54506 by kwonjun1202 last updated on 05/Feb/19 $${L}'{Hopital}\:{rule} \\ $$$$\underset{{x}\rightarrow\alpha} {\mathrm{lim}}\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}=\underset{{x}\rightarrow\alpha} {\mathrm{lim}}\:\frac{{f}\:'\left({x}\right)}{{g}'\left({x}\right)}=\:\frac{{f}\:'\left(\alpha\right)}{{g}'\left(\alpha\right)} \\ $$$${f}\:'\left({x}\right)=\frac{{d}}{{dx}}{f}\left({x}\right)\:,\:{g}'\left({x}\right)=\frac{{d}}{{dx}}{g}\left({x}\right)\:{differential} \\ $$$$\left.{What}\:{is}\:{it}?\:{Proof}\:{of}\:{the}\:{rule}..\:\mathrm{plz}\::\right) \\ $$ Answered by kaivan.ahmadi last updated…
Question Number 185537 by mathlove last updated on 23/Jan/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{\mathrm{2}{x}} −\mathrm{4}{e}^{{x}} +\mathrm{2}{x}+\mathrm{3}}{{x}^{\mathrm{3}} }=? \\ $$ Commented by Ar Brandon last updated on 23/Jan/23 $$\mathrm{Same}\:\mathrm{method}\:\mathrm{as}\:\mathrm{above}…
Question Number 185539 by mathlove last updated on 23/Jan/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{4}{e}^{\mathrm{3}{x}} −\mathrm{9}{e}^{\mathrm{2}{x}} +\mathrm{6}{x}+\mathrm{5}}{{x}^{\mathrm{3}} }=? \\ $$ Answered by Ar Brandon last updated on 23/Jan/23 $$\mathscr{L}=\underset{{x}\rightarrow\mathrm{0}}…
Question Number 185508 by mathlove last updated on 23/Jan/23 $$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\mathrm{5}^{{x}} −\mathrm{25}}{{x}−\mathrm{2}}=? \\ $$$${with}\:{out}\:{H}'{L}\:{roule} \\ $$ Answered by Ar Brandon last updated on 23/Jan/23 $$\underset{{x}\rightarrow\mathrm{2}}…