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 $$\underset{{x}\rightarrow\mathrm{2}}…
Question Number 119977 by bramlexs22 last updated on 28/Oct/20 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\sqrt[{\mathrm{5}}]{{x}^{\mathrm{4}} }\left(\sqrt[{\mathrm{5}}]{{x}+\mathrm{1}}−\sqrt[{\mathrm{5}}]{{x}}\:\right)\right)=? \\ $$ Answered by bemath last updated on 28/Oct/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\mathrm{5}}]{{x}^{\mathrm{5}} +{x}^{\mathrm{4}} }−\sqrt[{\mathrm{5}}]{{x}^{\mathrm{5}}…
Question Number 185510 by mathlove last updated on 23/Jan/23 Answered by mahdipoor last updated on 23/Jan/23 $$\mathrm{6}{x}−\mathrm{8}{sin}\left({x}\right)+{sin}\left(\mathrm{2}{x}\right)= \\ $$$$\mathrm{6}{x}−\mathrm{8}\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−\frac{{x}^{\mathrm{7}} }{\mathrm{7}!}+….\right)+ \\ $$$$\left(\mathrm{2}{x}−\frac{\mathrm{8}{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{\mathrm{32}{x}^{\mathrm{5}}…
Question Number 119961 by bobhans last updated on 28/Oct/20 $${Without}\:{L}'{Hopital}\:{rule}\: \\ $$$$\:\underset{{x}\rightarrow\pi/\mathrm{3}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left({x}−\frac{\pi}{\mathrm{3}}\right)}{\mathrm{1}−\mathrm{2cos}\:{x}}\:? \\ $$ Answered by bramlexs22 last updated on 28/Oct/20 Answered by Dwaipayan…
Question Number 54409 by pooja24 last updated on 03/Feb/19 $${li}\underset{{x}\rightarrow\infty} {{m}}\sqrt{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)}−{x}=? \\ $$$${pls}\:{solve}\:{this} \\ $$$$ \\ $$ Commented by maxmathsup by imad last updated…
Question Number 119902 by benjo_mathlover last updated on 28/Oct/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{cos}\:{x}\right)^{\frac{\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} {x}}} \:? \\ $$ Answered by benjo_mathlover last updated on 28/Oct/20 Answered by Olaf…