Question Number 203668 by Perelman last updated on 25/Jan/24 Answered by witcher3 last updated on 25/Jan/24 $$\mathrm{x}\overset{\mathrm{f}} {\rightarrow}\frac{\mathrm{1}}{\mathrm{4}−\mathrm{3x}} \\ $$$$\mathrm{f}'\left(\mathrm{x}\right)=\frac{\mathrm{3}}{\left(\mathrm{4}−\mathrm{3x}\right)^{\mathrm{2}} } \\ $$$$\forall\mathrm{x}\in\left[−\mathrm{1},\frac{\mathrm{1}}{\mathrm{3}}\right]\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\in\left[\frac{\mathrm{1}}{\mathrm{7}},\frac{\mathrm{1}}{\mathrm{3}}\right]…..\left(\mathrm{1}\right) \\ $$$$\left.\Rightarrow\mathrm{f}\left[−\mathrm{1},\frac{\mathrm{1}}{\mathrm{3}}\right]\right)\subset\left[\frac{\mathrm{1}}{\mathrm{7}},\frac{\mathrm{1}}{\mathrm{3}}\right]…
Question Number 203508 by Fridunatjan08 last updated on 20/Jan/24 $${Evaluate}\:{the}\:{given}\:{limit}: \\ $$$$\underset{{n}\rightarrow\infty} {{lim}}\frac{\sqrt[{{n}}]{\mathrm{8}}−\mathrm{1}}{\:\sqrt[{{n}}]{\mathrm{16}}−\mathrm{1}} \\ $$ Answered by esmaeil last updated on 20/Jan/24 $$={Y} \\ $$$$\frac{\mathrm{1}}{{n}}={p}\rightarrow\left({n}\rightarrow\infty\rightarrow{p}\rightarrow\mathrm{0}\right)…
Question Number 203330 by Calculusboy last updated on 16/Jan/24 Answered by MM42 last updated on 16/Jan/24 $$={lim}_{{x}\rightarrow\mathrm{1}} \:\frac{{sin}\left({x}−\mathrm{1}\right)}{\left({x}−\mathrm{1}\right)}\:×\frac{\mathrm{1}}{{x}+\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\checkmark \\ $$ Commented by Calculusboy last updated…
Question Number 203247 by mathlove last updated on 13/Jan/24 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}\:{tan}\frac{\pi}{\mathrm{2}}\left(\mathrm{1}+{x}\right)=? \\ $$ Answered by MM42 last updated on 13/Jan/24 $$={lim}_{{x}\rightarrow\mathrm{0}} \:−{xcot}\frac{\pi}{\mathrm{2}}{x}={lim}_{{x}\rightarrow\mathrm{0}} \:−\frac{\frac{\pi}{\mathrm{2}}{xcos}\frac{\pi}{\mathrm{2}}{x}}{{sin}\frac{\pi}{\mathrm{2}}{x}}×\frac{\mathrm{2}}{\pi} \\ $$$$=\:−\frac{\mathrm{2}}{\pi}\:\checkmark…
Question Number 203180 by Calculusboy last updated on 11/Jan/24 Answered by Rana_Ranino last updated on 11/Jan/24 $$\mathrm{using}\:\mathrm{arcsin}^{\mathrm{2}} \left(\mathrm{z}\right)=\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{4}^{\mathrm{n}} \mathrm{z}^{\mathrm{2n}} }{\mathrm{n}^{\mathrm{2}} \begin{pmatrix}{\mathrm{2n}}\\{\:\mathrm{n}}\end{pmatrix}}\:\:\mathrm{take}\:\mathrm{z}=\frac{\mathrm{1}}{\mathrm{2}}\: \\ $$$$\underset{\mathrm{n}=\mathrm{1}}…
Question Number 203059 by hassanmpsy last updated on 08/Jan/24 Commented by witcher3 last updated on 11/Jan/24 $$\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)}…