This is the application of the Lobida rule, which is directly derived from the numerator and denominator.

(x^n)'=nx^(n- 1)

[e^(λx)]'=λe^(λx]

∴lim(x->； +∞)[x^n/e^(λx]

= lim(x->； +∞)【x^n]'/[e^(λx)]'

= lim(x->； +∞)[nx^(n- 1)/[λe^(λx]

= lim(x->； +∞)【n(n- 1)x^(n-2)/[λ？ E (λ x)] (continue to deduce)

......

= lim(x->； +∞) [n(n- 1)(n-2)*...*2* 1/[λ^n*e^(λx)]

= lim(x->； +∞) (n！ )/[λ^n*e^(λx)]

=(n！ /λ^n)lim(x->； +∞) 1/[e^(λx)]

=(n！ /λ^n)*0

=0