`lim_(x->oo)(x^2-x^3)e^(2x)`

`=lim_(x->oo)x^3(1/x-1)e^(2x)`

Apply the Algebraic limit theorem,

`=lim_(x->oo)x^3(lim_(x->oo)1/x-1)lim_(x->oo)e^(2x)`

plug in the value of x

`=oo(0-1)oo`

`=-oo`

`lim_(x->oo)(x^2-x^3)e^(2x)`

`=lim_(x->oo)x^3(1/x-1)e^(2x)`

Apply the Algebraic limit theorem,

`=lim_(x->oo)x^3(lim_(x->oo)1/x-1)lim_(x->oo)e^(2x)`

plug in the value of x

`=oo(0-1)oo`

`=-oo`