Question Number 112851 by bemath last updated on 10/Sep/20 $$\mathrm{Without}\:\mathrm{L}'\mathrm{Hopital} \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{a}}{\mathrm{x}}\:−\:\mathrm{cot}\:\frac{\mathrm{x}}{\mathrm{a}}\right)\:? \\ $$ Answered by bobhans last updated on 10/Sep/20 $$\:\mathrm{set}\:\frac{\mathrm{x}}{\mathrm{a}}\:=\:\mathrm{t}\:\rightarrow\frac{\mathrm{a}}{\mathrm{x}}\:=\:\frac{\mathrm{1}}{\mathrm{t}} \\ $$$$\:\mathrm{L}=\:\underset{\mathrm{t}\rightarrow\mathrm{0}}…
Question Number 112782 by bemath last updated on 09/Sep/20 $$\mathrm{without}\:\mathrm{L}'\mathrm{Hopital} \\ $$$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}{\mathrm{x}−\pi} \\ $$ Answered by john santu last updated on 09/Sep/20 $${setting}\:{x}\:=\:\pi+{s}\:,\:{s}\rightarrow\mathrm{0} \\…
Question Number 47174 by vajpaithegrate@gmail.com last updated on 05/Nov/18 $$\mathrm{y}=\mathrm{log}_{\mathrm{2}} \left(\mathrm{log}_{\mathrm{2}} ^{\mathrm{x}} \right)\mathrm{then}\:\:\frac{\mathrm{dy}}{\mathrm{dx}}= \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 05/Nov/18 $${y}={log}_{\mathrm{2}} {t}=\frac{{lnt}}{{ln}\mathrm{2}} \\…
Question Number 112651 by bemath last updated on 09/Sep/20 $$\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\left(\mathrm{x}−\mathrm{a}\right)\:\mathrm{cosec}\:\left(\frac{\pi\mathrm{x}}{\mathrm{a}}\right)\:? \\ $$$$\left(\mathrm{2}\right)\:\int\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{dx}\:,\:\mathrm{by}\:\mathrm{using}\:\mathrm{Euler}'\mathrm{s} \\ $$$$\mathrm{substitution} \\ $$ Answered by john santu last updated on…
Question Number 112650 by bemath last updated on 09/Sep/20 $$\mathrm{Determine}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\mathrm{and}\:\mathrm{e}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{ax}+\mathrm{bx}^{\mathrm{3}} +\mathrm{cx}^{\mathrm{2}} +\mathrm{dx}+\mathrm{e}}{\mathrm{x}^{\mathrm{4}} }\:=\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{find}\:\mathrm{general}\:\mathrm{solution}\::\:\mathrm{3xy}^{\mathrm{2}} \:\mathrm{y}'\:=\:\mathrm{4y}^{\mathrm{3}} −\mathrm{x}^{\mathrm{3}} \: \\ $$ Answered by…
Question Number 178178 by cortano1 last updated on 13/Oct/22 Answered by Frix last updated on 14/Oct/22 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−\mathrm{sin}^{−\mathrm{1}} \:{x}}{{x}^{\mathrm{2}} \mathrm{sin}^{−\mathrm{1}} \:{x}}\:=_{\left[{t}=\mathrm{sin}^{−\mathrm{1}} \:{x}\right]} \\ $$$$=−\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{t}−\mathrm{sin}\:{t}}{{t}\mathrm{sin}^{\mathrm{2}}…
Question Number 112637 by bemath last updated on 09/Sep/20 $$\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{sinh}\:\mathrm{x}}{\mathrm{x}}\right)^{\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }} ? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{x}\:\mathrm{ln}\:\left(\mathrm{tan}\:\mathrm{x}\right)\:? \\ $$ Answered by john santu last updated on…
Question Number 178161 by mathlove last updated on 13/Oct/22 Commented by CElcedricjunior last updated on 15/Oct/22 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\boldsymbol{\mathrm{cosx}}−\boldsymbol{\mathrm{cos}}\mathrm{3}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{ax}}^{\mathrm{2}} }=\frac{\mathrm{0}}{\mathrm{0}}=\boldsymbol{\mathrm{FI}} \\ $$$$\boldsymbol{{to}}\:\boldsymbol{{apply}}\:\boldsymbol{{hospital}}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\boldsymbol{{sinx}}}{\boldsymbol{{a}}\mathrm{2}\boldsymbol{{x}}}+\frac{\mathrm{3}\boldsymbol{{sin}}\mathrm{3}\boldsymbol{{x}}}{\mathrm{2}\boldsymbol{{ax}}}+\frac{\boldsymbol{{sin}}\mathrm{2}\boldsymbol{{x}}}{\mathrm{2}\boldsymbol{{ax}}}…
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Question Number 47070 by Meritguide1234 last updated on 04/Nov/18 Commented by maxmathsup by imad last updated on 04/Nov/18 $${let}\:\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{cos}\left({nx}\right)}{{n}!}\:\Rightarrow{S}\left({x}\right)\:={Re}\left(\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{e}^{{inx}} }{{n}!}\right)\:{but} \\…