Question Number 203059 by hassanmpsy last updated on 08/Jan/24 Commented by witcher3 last updated on 11/Jan/24 $$\mathrm{U}_{\mathrm{n}} =\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{n}\left(\mathrm{1}+\frac{\mathrm{k}}{\mathrm{n}}\right)}{\mathrm{n}^{\mathrm{2}} \left(\mathrm{2}+\mathrm{2}\frac{\mathrm{k}}{\mathrm{n}}+\left(\frac{\mathrm{k}}{\mathrm{n}}\right)^{\mathrm{2}} \right)}=\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{f}\left(\frac{\mathrm{k}}{\mathrm{n}}\right) \\…
Question Number 202530 by Calculusboy last updated on 28/Dec/23 Answered by MathematicalUser2357 last updated on 29/Dec/23 $$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{log}\left(\mathrm{1}+{x}\right)^{\mathrm{1}+{x}} −{x}}{{x}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\partial^{\mathrm{2}} }{\partial^{\mathrm{2}} {x}}\left(\mathrm{log}\left(\mathrm{1}+{x}\right)^{\mathrm{1}+{x}}…
Question Number 202249 by cortano12 last updated on 23/Dec/23 $$\:\:\:\:\mathrm{C}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}}\:−\:\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{5}} }\:=? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 202160 by Calculusboy last updated on 22/Dec/23 Answered by MathematicalUser2357 last updated on 22/Dec/23 $$\frac{\mathrm{1}}{\mathrm{7}!}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{d}^{\mathrm{7}} }{{dx}^{\mathrm{7}} }\:\left(\mathrm{sin}\left(\mathrm{tan}\:{x}\right)−\mathrm{tan}\left(\mathrm{sin}\:{x}\right)\right) \\ $$ Terms of Service…
Question Number 202161 by Calculusboy last updated on 22/Dec/23 Answered by som(math1967) last updated on 22/Dec/23 $$\frac{\left({m}+\mathrm{1}\right)!\left(\mathrm{1}+\mathrm{3}+\mathrm{5}+…+\mathrm{2}{m}+\mathrm{3}\right.}{\mathrm{2}{m}\left({m}+\mathrm{2}\right)\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+{m}+\mathrm{1}\right)} \\ $$$$=\frac{\left({m}+\mathrm{1}\right)!\frac{{m}+\mathrm{2}}{\mathrm{2}}×\mathrm{2}\left\{\mathrm{1}+\left({m}+\mathrm{2}−\mathrm{1}\right)\right\}}{\mathrm{2}{m}\left({m}+\mathrm{2}\right)\frac{\left({m}+\mathrm{1}\right)\left({m}+\mathrm{2}\right)}{\mathrm{2}}} \\ $$$$=\frac{\left({m}+\mathrm{1}\right)!\left({m}+\mathrm{2}\right)^{\mathrm{2}} }{{m}\left({m}+\mathrm{2}\right)^{\mathrm{2}} \left({m}+\mathrm{1}\right)} \\ $$$$=\frac{{m}\left({m}+\mathrm{1}\right)×\left({m}−\mathrm{1}\right)!}{{m}\left({m}+\mathrm{1}\right)}…
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Question Number 201724 by LimPorly last updated on 11/Dec/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{{x}} \left({x}−\mathrm{2}\right)+{x}+\mathrm{2}}{{x}^{\mathrm{3}} }\:{solve}\:{it}\:{by}\:{not}\:{using} \\ $$$${taylor}\:{series}\:{or}\:{l}'{hopital}\:{rule}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 201657 by LimPorly last updated on 10/Dec/23 $${if}\:\:{f}\left({x}\right)=\begin{cases}{\frac{\mathrm{sin}\:\left(\mathrm{1}+\left[{x}\right]\right)}{\left[{x}\right]}\:\:{for}\:\left[{x}\right]\neq\mathrm{0}}\\{\mathrm{0}\:\:{for}\:\left[{x}\right]=\mathrm{0}}\end{cases} \\ $$$${where}\:\left[{x}\right]\:{represents}\:{an}\:{integer}\:\boldsymbol{{x}}\:{greatest}\:\leqslant\:\boldsymbol{{x}} \\ $$$${Find}\:\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}{f}\left({x}\right). \\ $$ Answered by aleks041103 last updated on 10/Dec/23…
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Question Number 201680 by cortano12 last updated on 10/Dec/23 $$\:\:\:\Subset \\ $$ Answered by Calculusboy last updated on 11/Dec/23 $$\boldsymbol{{Solution}}:\:\boldsymbol{{substitute}}\:\boldsymbol{{ditectly}},\:\boldsymbol{{we}}\:\boldsymbol{{get}}\:\frac{\mathrm{0}}{\mathrm{0}}\left(\boldsymbol{{indeterminant}}\right) \\ $$$$\boldsymbol{{let}}\:\boldsymbol{{p}}=\mathrm{2}\boldsymbol{{sinx}}−\boldsymbol{{sim}}\mathrm{2}\boldsymbol{{x}}\:\:\:\:\:\frac{\boldsymbol{{dp}}}{\boldsymbol{{dx}}}=\mathrm{2}\boldsymbol{{cosx}}−\mathrm{2}\boldsymbol{{cos}}\mathrm{2}\boldsymbol{{x}} \\ $$$$\boldsymbol{{let}}\:\boldsymbol{{q}}=\boldsymbol{{sinx}}−\boldsymbol{{xcosx}}\:\:\:\frac{\boldsymbol{{dq}}}{\boldsymbol{{dx}}}=\boldsymbol{{cosx}}−\left(\boldsymbol{{cosx}}−\boldsymbol{{xsinx}}\right) \\…