Question Number 73565 by Rio Michael last updated on 13/Nov/19 $${investigate}\:{the}\:{continuity}\:{of}\:{f}\:,{given}\:{by} \\ $$$${f}:\:{x}\:\rightarrow\:\begin{cases}{\mathrm{1}−{x},\:{if}\:{x}<\mathrm{1}}\\{\mathrm{0},{if}\:{x}\:=\mathrm{1}}\\{{x}^{\mathrm{2}} −\mathrm{3}{x}\:+\:\mathrm{2},{if}\:{x}\:>\mathrm{1}}\end{cases} \\ $$$${at}\:{the}\:{point}\:{x}\:=\mathrm{1} \\ $$$$ \\ $$ Commented by Rio Michael last…
Question Number 73561 by Rio Michael last updated on 13/Nov/19 $${find}\:{the}\:{value}\:{of}\:\lambda\:{for}\:{which} \\ $$$$\:{f}\::\:{x}\:\rightarrow\:\begin{cases}{\mathrm{2}\lambda\:−\:{x},\:{if}\:{x}\:<\:\mathrm{1}}\\{\lambda^{\mathrm{2}} \:+\:{x}\:−\mathrm{1},\:{if}\:{x}\:>\:\mathrm{1}}\end{cases} \\ $$$${has}\:{a}\:{limit}\:{as}\:{x}\rightarrow\:\mathrm{1} \\ $$ Answered by ajfour last updated on 13/Nov/19…
Question Number 73555 by Rio Michael last updated on 13/Nov/19 $${find}\: \\ $$$$\left.{a}\right)\:\:\underset{{x}\rightarrow−\infty} {{Lim}}\:\frac{{ln}\left(\mathrm{1}+{e}^{{x}} \right)}{{e}^{{x}} } \\ $$$$\left.{b}\right)\:\underset{{x}\rightarrow+\infty} {{lim}}\:\left(\sqrt{{x}^{\mathrm{2}} \:+\:\mathrm{3}{x}}\:\:−{x}\right) \\ $$$$\left.{c}\right)\:\underset{{x}\rightarrow\mathrm{0}} {{lim}}\:\sqrt{{x}}\:{ln}\left({sinx}\right) \\ $$$$\left.{d}\right)\:\:\underset{{x}\rightarrow+\infty}…
Question Number 73552 by Rio Michael last updated on 13/Nov/19 $${evaluate}\: \\ $$$$\underset{{x}\rightarrow+\infty} {\:{Lim}}\:{xln}\:\left(\frac{{x}+\mathrm{1}}{{x}}\right) \\ $$ Commented by mathmax by abdo last updated on 13/Nov/19…
Question Number 139059 by sahnaz last updated on 21/Apr/21 Answered by mr W last updated on 21/Apr/21 $$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{2}×\left(\frac{\mathrm{2}}{\mathrm{4}}\right)^{{n}} +\mathrm{9}×\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{{n}} +\mathrm{64}}{\left(\frac{\mathrm{2}}{\mathrm{4}}\right)^{{n}} +\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{{n}} +\mathrm{1}} \\ $$$$=\frac{\mathrm{2}×\mathrm{0}+\mathrm{9}×\mathrm{0}+\mathrm{64}}{\mathrm{0}+\mathrm{0}+\mathrm{1}}…
Question Number 73518 by aliesam last updated on 13/Nov/19 Commented by mathmax by abdo last updated on 13/Nov/19 $${let}\:{A}\left({x}\right)=\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right)^{\mathrm{2}{x}−\mathrm{1}} \:\Rightarrow{A}\left({x}\right)={e}^{\left(\mathrm{2}{x}−\mathrm{1}\right){ln}\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right)} \\ $$$$={e}^{\left(\mathrm{2}{x}−\mathrm{1}\right){ln}\left(\frac{{x}−\mathrm{1}+\mathrm{2}}{{x}−\mathrm{1}}\right)} \:={e}^{\left(\mathrm{2}{x}−\mathrm{1}\right){ln}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}−\mathrm{1}}\right)} \:\:{we}\:{have}\:{ln}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}−\mathrm{1}}\right)\sim\frac{\mathrm{2}}{{x}−\mathrm{1}}\left({x}\rightarrow+\infty\right) \\…
Question Number 139001 by bramlexs22 last updated on 21/Apr/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{1}−\mathrm{2tan}\:\mathrm{x}}}{\mathrm{sin}\:\mathrm{x}+\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}}\:=? \\ $$ Answered by EDWIN88 last updated on 21/Apr/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{x}\left(\frac{\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}}…
Question Number 138996 by bramlexs22 last updated on 21/Apr/21 $$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}}−\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}}{\:\sqrt{\mathrm{2}}\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}\:=? \\ $$$$ \\ $$ Answered by EDWIN88 last updated on 21/Apr/21 $$\:\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}{\:\sqrt{\mathrm{2}}\left(\mathrm{1}−\mathrm{sin}\:\mathrm{x}\right)}.\underset{{x}\rightarrow\pi/\mathrm{2}}…
Question Number 138997 by bramlexs22 last updated on 21/Apr/21 $$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\sqrt[{\mathrm{3}}]{\mathrm{7}+\mathrm{x}^{\mathrm{3}} }−\sqrt{\mathrm{3}+\mathrm{x}^{\mathrm{2}} }}{\mathrm{x}−\mathrm{1}}=? \\ $$ Answered by EDWIN88 last updated on 21/Apr/21 $$\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{7}+\mathrm{x}^{\mathrm{3}}…
Question Number 138959 by mathlove last updated on 20/Apr/21 Answered by mathmax by abdo last updated on 20/Apr/21 $$\mathrm{a}^{\mathrm{x}^{\mathrm{2}} −\mathrm{1}} \:=\mathrm{e}^{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\mathrm{loga}} \:\sim\mathrm{1}+\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\mathrm{loga}\:\mathrm{and} \\…