Question Number 217626 by mnjuly1970 last updated on 17/Mar/25
$$ \\ $$$$\:\:\:\:\:\:\mathrm{lim}_{\:\lambda\rightarrow\mathrm{0}} \:\int_{\lambda} ^{\:\mathrm{2}\lambda} \:\frac{\:{e}^{\mathrm{2}{t}\:} }{{t}}\:{dt}\:=\:? \\ $$$$ \\ $$
Answered by maths2 last updated on 17/Mar/25
$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\int_{{x}} ^{\mathrm{2}{x}} \frac{{e}^{\mathrm{2}{t}} −\mathrm{1}}{{t}}+\frac{\mathrm{1}}{{t}}{dt} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left\{\int_{{x}} ^{\mathrm{2}{x}} \frac{{e}^{\mathrm{2}{t}} −\mathrm{1}}{{t}}{dt}+\int_{{x}} ^{\mathrm{2}{x}} \frac{\mathrm{1}}{{t}}\right\}{dt} \\ $$$$={ln}\left(\mathrm{2}\right)+\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\int_{{x}} ^{\mathrm{2}{x}} \frac{{e}^{\mathrm{2}{t}} −\mathrm{1}}{{t}}{dt}_{=\mathrm{0}} \\ $$$${t}\overset{\overset{\ast} {{f}}} {\rightarrow}\frac{{e}^{\mathrm{2}{t}} −\mathrm{1}}{{t}}\:\:{can}\:{bee}\:{defind}\:{as}\:{contins}\:{function}\:{over} \\ $$$$\mathbb{R}\:\overset{\ast} {{f}\begin{cases}{{f}\:{if}\:{x}#\mathrm{0}}\\{\mathrm{2}\:{if}\:{x}=\mathrm{0}}\end{cases}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\int_{{x}} ^{\mathrm{2}{x}} \frac{{e}^{\mathrm{2}{g}} }{{g}}{dg}={ln}\left(\mathrm{2}\right) \\ $$
Commented by mnjuly1970 last updated on 17/Mar/25
$${thanks}\:{alot}\:{sir}\:. \\ $$
Commented by maths2 last updated on 17/Mar/25
$${Withe}\:{Pleasur}\:{God}\:{Bless}\:{You} \\ $$