Question Number 194451 by horsebrand11 last updated on 07/Jul/23 $$\:\:\:\:\:\:\cancel{\underline{\underbrace{\Vvdash}}} \\ $$ Answered by cortano12 last updated on 07/Jul/23 $$\: \\ $$ Commented by horsebrand11…
Question Number 194429 by horsebrand11 last updated on 06/Jul/23 $$\:\:\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\begin{cases}{\mathrm{x}^{\mathrm{2}} −\mathrm{5x}+\mathrm{a}\:;\:\mathrm{x}>\mathrm{1}}\\{\frac{\mathrm{3x}+\mathrm{2}}{\mathrm{x}−\mathrm{b}}\:;\:\mathrm{x}\leqslant\mathrm{1}}\end{cases} \\ $$$$\:\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{passes}\:\mathrm{through}\:\mathrm{at}\:\mathrm{point}\: \\ $$$$\:\:\left(\mathrm{2},−\mathrm{4}\right)\:\mathrm{and}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{exist}\:,\:\mathrm{find} \\ $$$$\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{3a}+\mathrm{2b}.\: \\ $$ Answered by MM42 last updated…
Question Number 194425 by cortano12 last updated on 06/Jul/23 $$\:\:\:\:\:\:\:\underbrace{ } \\ $$ Answered by horsebrand11 last updated on 06/Jul/23 $$\:\:\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\sqrt{\mathrm{sin}\:\mathrm{x}}\:\left(\mathrm{1}+\frac{\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{cos}\:\mathrm{x}}\right)}{\:\sqrt{\mathrm{x}}\:\left(\mathrm{1}+\:\sqrt{\mathrm{x}}\:\right)}=\:\mathrm{1} \\ $$…
Question Number 194335 by mathlove last updated on 04/Jul/23 Answered by cortano12 last updated on 04/Jul/23 $$\:\:\:\underbrace{\lesseqgtr^{} } \\ $$ Commented by mathlove last updated…
Question Number 194334 by cortano12 last updated on 04/Jul/23 $$\:\:\:\:\underbrace{ ^{} } \\ $$ Commented by Frix last updated on 04/Jul/23 $$\mathrm{The}\:\mathrm{real}\:\mathrm{part}\:\rightarrow\:\mathrm{0}\:\mathrm{but}\:\mathrm{the}\:\mathrm{imaginary}\:\mathrm{part} \\ $$$$\mathrm{is}\:<\mathrm{0}\:\mathrm{for}\:{x}<\mathrm{0}\:\mathrm{and}\:>\mathrm{0}\:\mathrm{for}\:{x}>\mathrm{0}\:\Rightarrow\:\mathrm{limit}\:\mathrm{does} \\…
Question Number 194344 by mathlove last updated on 04/Jul/23 Answered by qaz last updated on 04/Jul/23 $$\mathrm{sin}\:^{\mathrm{2}} {x}=\left({x}−\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} +…\right)^{\mathrm{2}} ={x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{4}} +… \\ $$$${e}^{{x}} +{e}^{−{x}}…
Question Number 194338 by mathlove last updated on 04/Jul/23 Answered by Peace last updated on 04/Jul/23 $${e}^{{cos}\left({t}\right){ln}\left(\mathrm{2}+{sin}\left({t}\right)\right)} −\mathrm{2}={f}\left({t}\right) \\ $$$$\left.{f}\:{in}\:\left[\mathrm{0},{x}\right];\exists{c}\in\right]\mathrm{0},{x}\left[\right. \\ $$$$\Rightarrow{f}\left({x}\right)−{f}\left(\mathrm{0}\right)={f}'\left({c}\right)\left({x}−\mathrm{0}\right) \\ $$$$\Rightarrow{f}'\left({c}\right)=\frac{\left(\mathrm{2}+{sin}\left({x}\right)\right)^{{cos}\left({x}\right)} −\mathrm{2}}{{x}}…
Question Number 194282 by alcohol last updated on 02/Jul/23 $${f}\left({f}\left({x}\right)\right)\:=\:{ax}\:+\:{b} \\ $$$$\mathrm{1}.\:{show}\:{that}\:{f}\left({ax}+{b}\right)\:=\:{af}\left({x}\right)\:+\:{b} \\ $$$${deduce}\:{f}\:'\left({ax}\:+\:{b}\right) \\ $$$$\mathrm{2}.\:{Show}\:{that}\:{f}\:'\left({x}\right)\:{is}\:{a}\:{constant}\: \\ $$$${hence}\:{deduce}\:{f} \\ $$ Answered by Frix last updated…
Question Number 194279 by cortano12 last updated on 02/Jul/23 $$\:\underbrace{ } \\ $$ Answered by horsebrand11 last updated on 02/Jul/23 $$\:\:\underline{\underbrace{ }\cancel{ }} \\ $$$$…
Question Number 194139 by cortano12 last updated on 28/Jun/23 Commented by MM42 last updated on 28/Jun/23 $${for} \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}^{\mathrm{2}} +\mathrm{2}{cosx}−\mathrm{2}}{{x}^{\mathrm{4}} }\:\rightarrow{hop} \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{2}\left({x}−{sinx}\right)}{\mathrm{4}{x}^{\mathrm{3}}…