Question Number 7200 by Tawakalitu. last updated on 16/Aug/16 $${Prove}\:{that}\: \\ $$$$\frac{{sin}\left(\frac{\Theta}{\mathrm{2}}\right)}{\left(\frac{\Theta}{\mathrm{2}}\right)}\:\:=\:\:\mathrm{1} \\ $$ Commented by Rasheed Soomro last updated on 16/Aug/16 $${Do}\:{you}\:{mean}\:\underset{\Theta\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\left(\frac{\Theta}{\mathrm{2}}\right)\:}{\frac{\Theta}{\mathrm{2}}}=\mathrm{1}\:? \\…
Question Number 72694 by MJS last updated on 31/Oct/19 $$\mathrm{convergent}\:\mathrm{or}\:\mathrm{divergent}? \\ $$$${S}=\frac{\mathrm{2}}{\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{1}}+\frac{\mathrm{2}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{2}}{\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{2}}{\mathrm{7}}−\frac{\mathrm{1}}{\mathrm{7}}… \\ $$ Commented by mathmax by abdo last updated on 31/Oct/19 $${S}=\sum_{{n}=\mathrm{0}} ^{\infty}…
Question Number 7101 by Tawakalitu. last updated on 10/Aug/16 Commented by Tawakalitu. last updated on 10/Aug/16 $${Working}\:{please}\:………… \\ $$ Answered by Yozzii last updated on…
Question Number 138153 by mathlove last updated on 10/Apr/21 $$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{{x}−\mid{tanx}\mid}{\mid{sinx}\mid−{x}}=? \\ $$ Answered by TheSupreme last updated on 10/Apr/21 $${for}\:{x}\rightarrow\mathrm{0}^{+} \:\mid{sin}\left({x}\right)\mid={sin}\left({x}\right)\:{and}\:\mid{tan}\left({x}\right)\mid={tan}\left({x}\right) \\ $$$${lim}\:\frac{{x}−{tan}\left({x}\right)}{{sin}\left({x}\right)−{x}}=…
Question Number 138140 by liberty last updated on 10/Apr/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)−\mathrm{cos}\:{x}}{{x}^{\mathrm{4}} }=? \\ $$ Answered by EDWIN88 last updated on 10/Apr/21 $$\:{recall}\:\mathrm{cos}\:{A}−\mathrm{cos}\:{B}\:=\:\mathrm{2sin}\:\left(\frac{{B}−{A}}{\mathrm{2}}\right)\mathrm{sin}\:\left(\frac{{A}+{B}}{\mathrm{2}}\right) \\ $$$${so}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2sin}\:\left(\frac{{x}−\mathrm{sin}\:{x}}{\mathrm{2}}\right)\mathrm{sin}\:\left(\frac{{x}+\mathrm{sin}\:{x}}{\mathrm{2}}\right)}{{x}^{\mathrm{4}}…
Question Number 6990 by Tawakalitu. last updated on 04/Aug/16 Answered by sou1618 last updated on 05/Aug/16 $${L}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left({x}\right)^{\mathrm{1}/{x}} \\ $$$${ln}\left({L}\right)=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}}{ln}\left({x}\right) \\ $$$${when}\:\mathrm{0}<{x} \\ $$$$\:\:\:\:\mathrm{0}<\frac{\mathrm{1}}{{x}},…
Question Number 72495 by Tony Lin last updated on 29/Oct/19 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{{lnx}}{{x}}\right)^{\frac{{lnx}}{{x}}} =? \\ $$ Commented by mathmax by abdo last updated on 29/Oct/19 $${let}\:{f}\left({x}\right)=\left(\frac{{lnx}}{{x}}\right)^{\frac{{lnx}}{{x}}}…
Question Number 6944 by Tawakalitu. last updated on 03/Aug/16 $${Prove}\:{that}: \\ $$$$ \\ $$$${e}^{{x}} =\:{Limit}\left({u}\:\rightarrow\:+\:\infty\right)\:\left(\mathrm{1}\:+\:\frac{{x}}{{u}}\right)^{{u}} \: \\ $$ Answered by sou1618 last updated on 03/Aug/16…
Question Number 6939 by sou1618 last updated on 03/Aug/16 $$\mathrm{please}\:\mathrm{solve}\:{L} \\ $$$$\mathrm{but}\:\mathrm{do}\:\mathrm{not}\:\mathrm{use}\:\mathrm{L}'\mathrm{Hopital}'\mathrm{s}\:\mathrm{rule}. \\ $$$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{{e}^{{x}} −\mathrm{1}}\right) \\ $$ Commented by Yozzii last updated on 03/Aug/16…
Question Number 72462 by 20190927 last updated on 29/Oct/19 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{4x}\:}\mathrm{cos}\left(\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{3}} \mathrm{arctan}\left(\mathrm{x}^{\mathrm{5}} \right)} \\ $$ Commented by kaivan.ahmadi last updated on 29/Oct/19 $${lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{4}{x}}\left(\mathrm{1}−\frac{{x}^{\mathrm{4}}…