Question Number 194685 by cortano12 last updated on 13/Jul/23 Answered by qaz last updated on 14/Jul/23 $$\left(\mathrm{1}+{x}\right)^{\frac{\mathrm{1}}{{x}}} ={e}^{\frac{\mathrm{1}}{{x}}{ln}\left(\mathrm{1}+{x}\right)} ={e}^{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{x}+\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{2}} +…} \\ $$$$={e}\left(\mathrm{1}+\left(−\frac{\mathrm{1}}{\mathrm{2}}{x}+\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{2}} +…\right)+\frac{\mathrm{1}}{\mathrm{2}}\left(−\frac{\mathrm{1}}{\mathrm{2}}{x}+\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{2}} +…\right)^{\mathrm{2}} +…\right.…
Question Number 194640 by cortano12 last updated on 12/Jul/23 $$\:\:\:\:\underbrace{ } \\ $$ Answered by horsebrand11 last updated on 12/Jul/23 $$\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}}}\:=\:\mathrm{y} \\ $$$$\:\underset{\mathrm{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}}{\mathrm{y}\:\mathrm{sin}\:\mathrm{3y}}\:=\underset{\mathrm{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}}…
Question Number 194462 by Erico last updated on 08/Jul/23 $$\mathrm{Prove}\:\mathrm{that}\:\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:+\infty^{} } \frac{\mathrm{1}−\boldsymbol{{e}}^{−\boldsymbol{{x}}^{\mathrm{2}} } }{\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}=\sqrt{\boldsymbol{\pi}} \\ $$ Answered by mnjuly1970 last updated on 08/Jul/23…
Question Number 194482 by horsebrand11 last updated on 08/Jul/23 $$\:\:\:\:\:\begin{array}{|c|}{\:\cancel{\underline{\underbrace{\Subset}}}}\\\hline\end{array} \\ $$ Answered by cortano12 last updated on 08/Jul/23 $$\:\:\:\:\mathrm{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left[\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\:\left(\frac{\mathrm{2}}{\mathrm{cos}\:\mathrm{x}}\:+\:\mathrm{cos}\:\mathrm{x}−\mathrm{3}\right)\right] \\ $$$$\:\:\:\mathrm{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{cos}\:^{\mathrm{2}}…
Question Number 194466 by horsebrand11 last updated on 08/Jul/23 $$\:\:\:\:\Subset \\ $$ Answered by cortano12 last updated on 08/Jul/23 $$\:\:\:\underbrace{\Subset} \\ $$ Answered by witcher3…
Question Number 194451 by horsebrand11 last updated on 07/Jul/23 $$\:\:\:\:\:\:\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} \\…