Question Number 76151 by Rio Michael last updated on 24/Dec/19 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{{k}} {e}^{−\mathrm{4}{x}} ,\:{k}>\mathrm{0} \\ $$ Commented by kaivan.ahmadi last updated on 24/Dec/19 $${lim}_{{x}−\rightarrow\infty} \frac{{x}^{{k}}…
Question Number 141627 by cherokeesay last updated on 21/May/21 Answered by MJS_new last updated on 21/May/21 $$\frac{\mathrm{sin}\:\mathrm{2}{x}\:+\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{1}+\mathrm{cos}\:\mathrm{2}{x}\:+\mathrm{cos}\:\mathrm{4}{x}}=\frac{−\mathrm{2sin}\:{x}\:\mathrm{cos}\:{x}\:\left(\mathrm{1}−\mathrm{4cos}^{\mathrm{2}} \:{x}\right)}{\mathrm{1}+\mathrm{2cos}^{\mathrm{2}} \:{x}\:\left(\mathrm{1}−\mathrm{4sin}^{\mathrm{2}} \:{x}\right)}= \\ $$$$\:\:\:\:\:\left[\mathrm{sin}\:{x}\:=\frac{\mathrm{tan}\:{x}}{\:\sqrt{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \:{x}}}\wedge\mathrm{cos}\:{x}\:=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \:{x}}}\right] \\…
Question Number 141509 by I want to learn more last updated on 19/May/21 $$\underset{\theta\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{sin}^{\mathrm{3}} \mathrm{3}\theta\:\:\:−\:\:\:\mathrm{3}\theta^{\mathrm{3}} }{\theta^{\mathrm{3}} \:\:+\:\:\theta^{\mathrm{4}} } \\ $$ Answered by bramlexs22 last…
Question Number 75939 by ahmadshahhimat775@gmail.com last updated on 21/Dec/19 Commented by JDamian last updated on 21/Dec/19 $$\left.{c}\right)\:\left({n}!\right)^{\frac{\mathrm{1}}{{n}}} \\ $$ Commented by mathmax by abdo last…
Question Number 75931 by Rio Michael last updated on 21/Dec/19 $${Evaluate} \\ $$$${a}.\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\sqrt[{\mathrm{3}}]{\frac{{x}^{\mathrm{2}} +\mathrm{3}}{\mathrm{27}{x}^{\mathrm{2}} −\mathrm{1}}} \\ $$$${b}.\:\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\frac{{x}−\mathrm{2}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$$${c}.\underset{{x}\rightarrow\infty} {\:\mathrm{lim}}\frac{{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{2}{x}−\mathrm{3}} \\…
Question Number 75930 by Rio Michael last updated on 21/Dec/19 $${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 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 $${lim}_{{x}\rightarrow−\infty}…
Question Number 141463 by bramlexs22 last updated on 19/May/21 $$\:\:\:\:\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 10374 by ridwan balatif last updated on 06/Feb/17 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{4}}} {\mathrm{lim}}\frac{\left(\mathrm{x}−\frac{\pi}{\mathrm{4}}\right)\mathrm{sin}\left(\mathrm{3x}−\mathrm{3}\frac{\pi}{\mathrm{4}}\right)}{\mathrm{2}\left(\mathrm{1}−\mathrm{sin2x}\right)}=…? \\ $$ Answered by mrW1 last updated on 06/Feb/17 $${let}\:{u}={x}−\frac{\pi}{\mathrm{4}} \\ $$$${with}\:{x}\rightarrow\frac{\pi}{\mathrm{4}},\:{u}\rightarrow\mathrm{0} \\…
Question Number 75743 by Master last updated on 16/Dec/19 Commented by Master last updated on 16/Dec/19 $$\mathrm{work}\:\mathrm{without}\:\mathrm{lopart} \\ $$ Commented by $@ty@m123 last updated on…