Question Number 134753 by mathocean1 last updated on 06/Mar/21
$${f}\:{is}\:{defined}\:{in}\:\left[\mathrm{0};\:+\infty\left[.\right.\right. \\ $$ $$\begin{cases}{\:}\\{}\\{{f}\left(\mathrm{0}\right)={ln}\mathrm{2}}\end{cases}{f}\left({x}\right)=\int_{{x}} ^{\mathrm{2}{x}} \:\frac{{e}^{−{t}} }{{t}}{dt}\:\:{for}\:{x}>\mathrm{0} \\ $$ $$ \\ $$ $$\left.\mathrm{1}\right)\:{Given}\:\mathrm{0}\leqslant{f}\left({x}\leqslant\frac{{e}^{−{x}} −{e}^{−\mathrm{2}{x}} }{{x}}.\right. \\ $$ $${Calcule}\:{the}\:{lim}\:{f}\left({x}\right)\:{at}\:\mathrm{0}\:{and}\:+\infty. \\ $$ $$\left.\mathrm{2}\right)\:{Calculate}\:{f}\:'\left({x}\right)\:,\:{give}\:{its}\:{variation} \\ $$ $${and}\:{plot}\:{its}\:{curve}. \\ $$ $$ \\ $$ $$ \\ $$