帮忙求极限,说明原因lim(x趋向正无穷)x^n*e^(-ax),(a>0,n为自然数)

Question

帮忙求极限,说明原因
lim(x趋向正无穷)x^n*e^(-ax),(a>0,n为自然数)

Answer 1

原式=lim(x->+∞)[x^n*e^(-ax)]
=lim(x->+∞)[x^n/e^(ax)]
=lim(x->+∞)[nx^(n-1)/(ae^(ax))] (∞/∞型极限,应用罗比达法则)
=lim(x->+∞)[n(n-1)x^(n-2)/(a²e^(ax))] (∞/∞型极限,应用罗比达法则)
.
=lim(x->+∞){n!/[(a^n)(e^(ax))]} (∞/∞型极限,应用罗比达法则)
=n!/(a^n)*lim(x->+∞)[1/e^(ax)]
=n!/(a^n)*0 (∵a>0)
=0