Illustration of bracketing

${\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)<{\rm {E}}_{1}(x)<\exp(-x)\!~\ln \!\left(1+{\frac {1}{x}}\right)~~~,~~~~~x\!>\!0}$

of the exponential integral ${\displaystyle ~{\rm {E}}_{1}(x)~}$ (black) with elementary functions ${\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)~}$ (blue) and ${\displaystyle ~\exp(-x)\!~\ln \!\left(1+{\frac {1}{x}}\right)~}$