Question Number 222829 by hardmath last updated on 08/Jul/25 $$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{3x}\:+\:\left[\mathrm{x}\right]}{\mathrm{2x}} \\ $$$$\mathrm{Find}\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow−\mathrm{5}^{+} } {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right)\:−\:\underset{\boldsymbol{\mathrm{x}}\rightarrow−\mathrm{5}^{−} } {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:? \\ $$ Answered by mehdee7396 last updated on 08/Jul/25…
Question Number 222830 by Tawa11 last updated on 08/Jul/25 Answered by Frix last updated on 08/Jul/25 $$\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}…= \\ $$$$=\frac{\mathrm{32}}{\mathrm{9}}\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{{i}^{\mathrm{3}} }{{k}^{\mathrm{4}} }+\mathrm{120}\underset{{i}=\mathrm{1}}…
Question Number 222811 by MrGaster last updated on 08/Jul/25 $$\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{2}{x}\right)}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$ Answered by MrGaster last updated on 08/Jul/25 Terms of Service…
Question Number 222805 by MrGaster last updated on 08/Jul/25 $$\mathrm{Prove}:\int_{−\infty} ^{\infty} {J}_{\mathrm{0}} \left(\mathrm{2}{x}\right){dx}=\mathrm{1} \\ $$ Answered by MrGaster last updated on 08/Jul/25 $${J}_{\mathrm{0}} \left(\mathrm{2}{x}\right)=\frac{\mathrm{1}}{\mathrm{2}\pi}\int_{−\pi} ^{\pi}…
Question Number 222781 by MathematicalUser2357 last updated on 07/Jul/25 $$\:\cancel{\underline{\underbrace{\succapprox}}} \\ $$ Answered by AntonCWX8 last updated on 07/Jul/25 $$ \\ $$ Commented by MathematicalUser2357…
Question Number 222783 by MathematicalUser2357 last updated on 07/Jul/25 $$\:\underbrace{ } \\ $$ Answered by AntonCWX8 last updated on 07/Jul/25 $${f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{5}}{x}^{\mathrm{5}} −\frac{\mathrm{5}}{\mathrm{3}}{x}^{\mathrm{3}} +\mathrm{4}{x}+\mathrm{2} \\ $$$${f}'\left({x}\right)={x}^{\mathrm{4}}…
Question Number 222777 by hardmath last updated on 07/Jul/25 $$\mathrm{Simplify}: \\ $$$$\frac{\left(\mathrm{cos214}°\:+\:\boldsymbol{\mathrm{i}}\:\mathrm{sin146}°\right)\centerdot\left(\mathrm{cos10}°\:+\:\boldsymbol{\mathrm{i}}\:\mathrm{sin10}°\right)}{\left(\mathrm{cos66}°\:−\:\boldsymbol{\mathrm{i}}\:\mathrm{sin246}°\right)}\:=\:? \\ $$ Answered by Frix last updated on 07/Jul/25 $$\mathrm{cos}\:\mathrm{214}°\:+\mathrm{i}\:\mathrm{sin}\:\mathrm{146}°\:=\mathrm{e}^{\mathrm{i}\frac{\mathrm{73}\pi}{\mathrm{90}}} \\ $$$$\mathrm{cos}\:\mathrm{10}°\:+\mathrm{i}\:\mathrm{sin}\:\mathrm{10}°\:=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{18}}} \\…
Question Number 222778 by Osefavour last updated on 07/Jul/25 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2log}\left(\mathrm{1}+\mathrm{x}\right)−\frac{\mathrm{x}\left(\mathrm{3x}+\mathrm{2}\right)}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }}{\mathrm{x}^{\mathrm{3}} } \\ $$ Answered by MrGaster last updated on 09/Jul/25 $$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2}\left({x}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}^{\mathrm{3}}…
Question Number 222779 by mnjuly1970 last updated on 07/Jul/25 Answered by gabthemathguy25 last updated on 08/Jul/25 $$\mathrm{this}\:\mathrm{is}\:\mathrm{actually}\:\mathrm{hard}.. \\ $$$$\mathrm{im}\:\mathrm{approximating}\:\mathrm{this}. \\ $$$$\approx−\mathrm{0}.\mathrm{0491713} \\ $$ Terms of…
Question Number 222800 by fantastic last updated on 07/Jul/25 Answered by mr W last updated on 07/Jul/25 $$\mathrm{2}×\frac{\pi{a}^{\mathrm{2}} }{\mathrm{6}}−\frac{\sqrt{\mathrm{3}}{a}^{\mathrm{2}} }{\mathrm{4}}=\left(\frac{\pi}{\mathrm{3}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}\right){a}^{\mathrm{2}} \approx\mathrm{0}.\mathrm{614}{a}^{\mathrm{2}} \\ $$ Commented by…