To determine

To evaluate the limit limx→−∞(x2−x3)e2x

Answer

  The value of the limit limx→−∞(x2−x3)e2x=0

Explanation

  The given limit is in indeterminate form

limx→−∞(x2−x3)e2x (∞×0 form)

Rewrite the limit

limx→−∞(x2−x3)e2x=limx→−∞(x2−x3)e−2x

Applying l’Hospitals Rule

limx→−∞(x2−x3)e2x=limx→−∞(2x−3x2)−2e−2x

Applying l’Hospitals Rule again, we get

limx→−∞(x2−x3)e2x=limx→−∞(2−6x)4e−2x

Applying l’Hospitals Rule again, we get

limx→−∞(x2−x3)e2x=limx→−∞−6−8e−2x=limx→−∞3e2x4=3(0)4=0

Conclusion:   The value of the limit limx→−∞(x2−x3)e2x=0