Question Number 121638 by bemath last updated on 10/Nov/20 $$\:\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{{xf}\left({x}\right)−{af}\left({a}\right)}{{x}−{a}}\:? \\ $$ Answered by Dwaipayan Shikari last updated on 10/Nov/20 $$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{xf}^{'} \left({x}\right)+{f}\left({x}\right)}{\mathrm{1}}=\:{af}'\left({a}\right)+{f}\left({a}\right) \\…
Question Number 121618 by SOMEDAVONG last updated on 10/Nov/20 Answered by bemath last updated on 10/Nov/20 $$\:\mathrm{let}\:\mathrm{cos}\:\mathrm{x}\:=\:\mathrm{u}^{\mathrm{mn}\:} \\ $$$$\:\underset{\mathrm{u}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{n}\left(\mathrm{1}−\mathrm{u}^{\mathrm{m}} \right)}\:−\:\frac{\mathrm{1}}{\mathrm{m}\left(\mathrm{1}−\mathrm{u}^{\mathrm{n}} \right)}\right)\:= \\ $$$$\:\underset{\mathrm{u}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\left(\frac{\mathrm{m}\left(\mathrm{1}−\mathrm{u}^{\mathrm{n}}…
Question Number 121576 by oustmuchiya@gmail.com last updated on 09/Nov/20 Answered by TANMAY PANACEA last updated on 09/Nov/20 $${f}\left({x}\right)=\frac{\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{15}}\:−\mathrm{5}\right)\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{15}}\:+\mathrm{5}\right)}{\left({x}−\mathrm{2}\right)\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{15}}\:+\mathrm{5}\right)} \\ $$$$=\frac{{x}^{\mathrm{2}} +\mathrm{15}−\mathrm{25}}{\left({x}−\mathrm{2}\right)\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{15}}\:+\mathrm{5}\right)}=\frac{{x}^{\mathrm{2}}…
Question Number 121574 by oustmuchiya@gmail.com last updated on 09/Nov/20 Answered by TANMAY PANACEA last updated on 09/Nov/20 $${t}={li}\underset{{n}\rightarrow\infty} {{m}}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \\ $$$${lnt}={li}\underset{{n}\rightarrow\infty} {{m}nln}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right) \\ $$$${lnt}={li}\underset{{y}\rightarrow\mathrm{0}} {{m}}\:\frac{{ln}\left(\mathrm{1}+{y}\right)}{{y}}=\mathrm{1}…
Question Number 121577 by oustmuchiya@gmail.com last updated on 09/Nov/20 Answered by TANMAY PANACEA last updated on 09/Nov/20 $${f}\left({x}\right)=\frac{\mathrm{3}+\frac{\mathrm{2}}{{x}}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}{\mathrm{1}−\frac{\mathrm{2}}{{x}^{\mathrm{2}} }} \\ $$$${li}\underset{{x}\rightarrow\infty} {{m}f}\left({x}\right)=\frac{\mathrm{3}+\mathrm{0}+\mathrm{0}}{\mathrm{1}}=\mathrm{3} \\ $$$$…
Question Number 121575 by oustmuchiya@gmail.com last updated on 09/Nov/20 Commented by Dwaipayan Shikari last updated on 09/Nov/20 $${R}−\left\{\mathrm{7},−\mathrm{7}\right\} \\ $$ Answered by TANMAY PANACEA last…
Question Number 187092 by cortano12 last updated on 13/Feb/23 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}^{\mathrm{3}} −\mathrm{sin}\:^{\mathrm{3}} {x}}{{x}^{\mathrm{3}} \left(\mathrm{cos}\:{x}^{\mathrm{3}} −\mathrm{cos}\:^{\mathrm{3}} {x}\right)}\:=? \\ $$ Answered by Farhadazizi last updated on 13/Feb/23…
Question Number 121527 by benjo_mathlover last updated on 09/Nov/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\pi+\mathrm{2sin}\:\mathrm{x}}\:−\sqrt{\pi}}\:? \\ $$ Answered by Dwaipayan Shikari last updated on 09/Nov/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{xcosx}}{\pi+\mathrm{2}{sinx}−\pi}\left(\sqrt{\pi+\mathrm{2}{sinx}}+\sqrt{\pi}\right) \\ $$$$=\frac{{x}}{\mathrm{2}{sinx}}\left(\sqrt{\pi}+\sqrt{\pi}\right)\:\:\:\:\:\:\:\:\:\left({sinx}\rightarrow{x}\:\:\:{and}\:{x}\rightarrow\mathrm{0}\right)…
Question Number 121506 by benjo_mathlover last updated on 08/Nov/20 $$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{x}+\mathrm{e}^{\frac{\mathrm{x}}{\mathrm{2}}} \right)^{\frac{\mathrm{2}}{\mathrm{x}}} \:?\: \\ $$ Answered by liberty last updated on 09/Nov/20 $$\mathrm{L}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{x}+\mathrm{e}^{\frac{\mathrm{x}}{\mathrm{2}}} \right)^{\frac{\mathrm{2}}{\mathrm{x}}}…
Question Number 121502 by bramlexs22 last updated on 08/Nov/20 $$\:\left(\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{2x}}}{\mathrm{x}\sqrt{\mathrm{2}}}\:? \\ $$$$\left(\mathrm{2}\right)\:\int\:\mathrm{cot}\:^{\mathrm{3}} \mathrm{x}\:\mathrm{csc}\:^{\mathrm{3}} \mathrm{x}\:\mathrm{dx}\: \\ $$$$ \\ $$ Commented by liberty last updated on…