Answer
$0$
Work Step by Step
$\lim _{x\rightarrow \infty }\dfrac {x^{3}}{e^{\dfrac {x}{2}}}=\dfrac {\dfrac {d}{dx}\left( x^{3}\right) }{\dfrac {d}{dx}\left( e^{\dfrac {x}{2}}\right) }=\dfrac {3x^{2}}{\dfrac {1}{2}e^{\dfrac {x}{2}}}=\dfrac {\dfrac {d}{dx}\left( 3x^{2}\right) }{\dfrac {d}{dx}\left( \dfrac {1}{2}e^{\dfrac {x}{2}}\right) }=\dfrac {6x}{\dfrac {1}{4}e^{\dfrac {x}{2}}}=\dfrac {\dfrac {d}{dx}\left( 6x\right) }{\dfrac {d}{dx}\left( \dfrac {1}{4}e^{\dfrac {x}{2}}\right) }=\dfrac {6}{\dfrac {1}{8}e^{\dfrac {x}{2}}}=\dfrac {6}{\infty }=0$
