Question Number 27787 by abdo imad last updated on 14/Jan/18 $${find}\:\:{lim}_{{x}−>\mathrm{0}} \frac{{e}^{{x}} \:\:−{x}−\mathrm{1}}{{x}^{\mathrm{2}} }\:. \\ $$ Answered by prakash jain last updated on 14/Jan/18 $${e}^{{x}}…
Question Number 158847 by mathlove last updated on 09/Nov/21 Answered by mr W last updated on 09/Nov/21 $$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{x}^{{x}} −{a}^{{a}} }{{x}−{a}}={a}^{{a}} \left(\mathrm{ln}\:{a}+\mathrm{1}\right) \\ $$$$\underset{{y}\rightarrow{x}} {\mathrm{lim}}\frac{{y}^{{y}}…
Question Number 93287 by john santu last updated on 12/May/20 Commented by mathmax by abdo last updated on 12/May/20 $${let}\:{f}\left({x}\right)\:=\frac{{arctan}\left({x}^{\mathrm{2}} \left(\mathrm{1}−{cosx}\right)\right)}{\left(\mathrm{1}−\sqrt{{cosx}}\right){ln}\left(\frac{{sinx}}{{x}}\right)}{we}\:{have}\:\mathrm{1}−{cosx}\:\sim\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow{x}^{\mathrm{2}} \left(\mathrm{1}−{cosx}\right)\sim\frac{{x}^{\mathrm{4}} }{\mathrm{2}}\:\Rightarrow \\…
Question Number 93258 by john santu last updated on 12/May/20 Commented by john santu last updated on 12/May/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3sin}\:\mathrm{x}−\mathrm{4sin}\:^{\mathrm{3}} \mathrm{x}+\mathrm{4sin}\:^{\mathrm{3}} \mathrm{x}−\mathrm{3ln}\left(\mathrm{1}+\mathrm{x}\right)}{\left(\mathrm{e}^{\mathrm{x}} −\mathrm{1}\right)\mathrm{sin}\:\mathrm{x}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}}…
Question Number 27727 by NECx last updated on 13/Jan/18 $${What}\:{are}\:{the}\:{conditions}\:{whereby} \\ $$$${the}\:{limit}\:{of}\:{a}\:{function}\:{does}\:{not} \\ $$$${exist}\:{at}\:{a}\:{poont}? \\ $$ Answered by prakash jain last updated on 13/Jan/18 $$\mathrm{When}\:\mathrm{LHL}\:\mathrm{is}\:\mathrm{not}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{RHL}.…
Question Number 27722 by NECx last updated on 13/Jan/18 $${find}\:{the}\:{limit}\:{of} \\ $$$$ \\ $$$${f}\left({x}\right)=\begin{cases}{\mathrm{1}+{x}\:\:\:\:\:\:\:{x}<\mathrm{1}}\\{{k}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}=\mathrm{0}\:\:\:{c}=\mathrm{0}}\\{\mathrm{1}+{x}\:\:\:\:\:,\:{x}>\mathrm{0}}\end{cases} \\ $$ Answered by NECx last updated on 15/Jan/18 $$\:\:{the}\:{limit}\:{is}\:{said}\:{to}\:{exist}\:{is}\:{the} \\…
Question Number 27723 by NECx last updated on 13/Jan/18 $$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{3}} −\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$ Answered by prakash jain last updated on 13/Jan/18 $$\frac{{x}^{\mathrm{3}} −\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}}…
Question Number 158780 by EbrimaDanjo last updated on 08/Nov/21 Answered by puissant last updated on 08/Nov/21 $${K}=\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{{e}^{{e}^{{e}^{{e}^{{x}} } } } } {e}^{{e}^{{e}^{{e}^{{x}} }…
Question Number 27701 by NECx last updated on 13/Jan/18 $${If}\:{the}\:{function}\:{f}\left({x}\right)\:{satisfies} \\ $$$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\:\frac{{f}\left({x}\right)−\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}}\:=\pi,\:{evaluate}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{f}\left({x}\right) \\ $$ Commented by abdo imad last updated on 21/Jan/18…
Question Number 27699 by NECx last updated on 13/Jan/18 $${suppose}\:{f}\left({x}\right)=\begin{cases}{{a}+{bx},\:\:{x}<\mathrm{1}}\\{\mathrm{4},\:\:\:\:\:\:\:{x}=\mathrm{1}}\\{{b}−{ax},\:\:{x}>\mathrm{1}}\end{cases}\:{and} \\ $$$${if}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:{f}\left({x}\right)={f}\left(\mathrm{1}\right)\:{what}\:{are}\:{possible} \\ $$$${values}\:{of}\:{a}\:{and}\:{b}? \\ $$$$ \\ $$ Answered by peileng8802 last updated on…