Question Number 83491 by john santu last updated on 03/Mar/20 $$\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\sqrt{\mathrm{1}+\mathrm{sin}\:\left(\mathrm{2x}\right)}−\sqrt{\mathrm{1}−\mathrm{2sin}\:\left(\mathrm{x}\right)}}{\mathrm{x}} \\ $$$$\mathrm{g}\left(\mathrm{x}\right)\:=\:\mathrm{2x}+\sqrt{\mathrm{2x}} \\ $$$$\mathrm{find}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\: \\ $$ Commented by john santu last updated on…
Question Number 148992 by mathlove last updated on 02/Aug/21 Answered by gsk2684 last updated on 02/Aug/21 $$\approx\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{sin}\:{x}+\frac{\mathrm{sin}\:^{\mathrm{2}} {x}}{\mathrm{2}!}+..\right)−\left(\mathrm{sin}\:\mathrm{3}{x}+\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{3}{x}}{\mathrm{2}!}+..\right)}{\mathrm{2}{x}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\left(\mathrm{1}+\mathrm{0}+…\right)−\left(\mathrm{3}+\mathrm{0}+..\right)\right\} \\ $$$$=−\mathrm{1} \\…
Question Number 148990 by gsk2684 last updated on 02/Aug/21 $${let}\:{f}:{R}\rightarrow{R}\:{be}\:{a}\:{continuius}\:{function} \\ $$$${such}\:{that}\:{for}\:{any}\:{two}\:{real}\:{numbers} \\ $$$${x}\:{and}\:{y}\:\mid{f}\left({x}\right)−{f}\left({y}\right)\mid\leqslant\mathrm{10}\mid{x}−{y}\mid^{\mathrm{201}} \\ $$$${then}\:{prove}\:{that} \\ $$$${f}\left(\mathrm{2019}\right)+{f}\left(\mathrm{2022}\right)=\mathrm{2}\:{f}\left(\mathrm{2021}\right) \\ $$ Terms of Service Privacy Policy…
Question Number 83401 by jagoll last updated on 02/Mar/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{6}−\mathrm{8xsin}\:\left(\frac{\mathrm{3}}{\mathrm{x}}\right)\right)=\:? \\ $$ Commented by mathmax by abdo last updated on 02/Mar/20 $$\mathrm{6}−\mathrm{8}{x}\:{sin}\left(\frac{\mathrm{3}}{{x}}\right)\sim\mathrm{6}−\mathrm{8}{x}×\frac{\mathrm{3}}{{x}}\:\Rightarrow{lim}_{{x}\rightarrow\infty} \left(\mathrm{6}−\mathrm{8}{x}\:{sin}\left(\frac{\mathrm{3}}{{x}}\right)\right)=\mathrm{6}−\mathrm{24}=−\mathrm{18} \\…
Question Number 83361 by john santu last updated on 01/Mar/20 $$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\left(\mathrm{x}\right)}{\:\sqrt{\mathrm{x}}}\:=\: \\ $$ Commented by john santu last updated on 01/Mar/20 $$\underset{{x}\rightarrow\mathrm{0}^{+} }…
Question Number 83329 by john santu last updated on 01/Mar/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{e}^{−\mathrm{2x}} −\left(\mathrm{1}+\mathrm{ax}\right)\right)}{\mathrm{x}^{\mathrm{2}} \:\left(\mathrm{1}+\mathrm{bx}\right)}\:=\:? \\ $$ Commented by jagoll last updated on 01/Mar/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{bx}}\:×\:\underset{{x}\rightarrow\mathrm{0}}…
Question Number 83323 by john santu last updated on 01/Mar/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\left(\mathrm{3x}−\mathrm{2}\right)\left(\mathrm{x}−\sqrt{\mathrm{2}}\right)}\:−\:\mathrm{x}\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\:=? \\ $$$$ \\ $$ Commented by abdomathmax last updated on 01/Mar/20 $${let}\:{f}\left({x}\right)=\sqrt{\left(\mathrm{3}{x}−\mathrm{2}\right)\left({x}−\sqrt{\mathrm{2}}\right)}−{x}\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}} \\…
Question Number 148849 by EDWIN88 last updated on 31/Jul/21 $$\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{{x}^{\mathrm{3}} −\mathrm{cos}\:\sqrt{{x}}\:\mathrm{sin}\:^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} }\:=? \\ $$ Answered by iloveisrael last updated on 01/Aug/21 $$\:\underset{{x}\rightarrow\mathrm{0}^{+}…
Question Number 148848 by EDWIN88 last updated on 31/Jul/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{4}}]{{x}+\mathrm{16}}\:\sqrt[{\mathrm{3}}]{{x}+\mathrm{8}}\:−\mathrm{4}}{{x}^{\mathrm{2}} +\mathrm{2}{x}}\:=?\: \\ $$ Answered by mathmax by abdo last updated on 01/Aug/21 $$\mathrm{f}\left(\mathrm{x}\right)=\frac{\left(\mathrm{x}+\mathrm{16}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} \left(\mathrm{x}+\mathrm{8}\right)^{\frac{\mathrm{1}}{\mathrm{3}}}…
Question Number 83287 by Power last updated on 29/Feb/20 Answered by mr W last updated on 29/Feb/20 $$\mathrm{sin}\:{x}={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−….. \\ $$$$\frac{\mathrm{sin}\:{x}}{{x}}=\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−….. \\…