Question

求这个函数的极限。

Answer 1

lim(x->+∞乱备 ) e^x/x^μ
case 1: μ是一个正整数
lim(x->+∞ ) e^x/x^μ (∞/∞分子分母分别求导)
=lim(x->+∞ ) e^x/[μx^(μ-1)] (∞/∞分子分母分别求导)
=lim(x->+∞ ) e^x/[μ(μ-1)x^(μ-2)] (∞/∞分子分母分别求导)
...
=lim(x->+∞ ) e^x/μ!
->+∞
case 1: μ不是一个正整数 ： μ =n+δ , n是一个正整数, 0<δ<1
lim(x->+∞ ) e^x/则陪腊x^μ (∞/∞分子分母分别求导)

=lim(x->+∞ ) e^x/[μx^(μ-1)] (∞/∞分子分母分别求导)
=lim(x->+∞ ) e^x/[μ(μ-1)x^(μ-2)] (∞/∞分子分母分别孙滑求导)
...
=lim(x->+∞ ) e^x/[μ(μ-1)(μ-2). (μ-n+1).x^δ ]
=lim(x->+∞ ) e^x/[μ(μ-1)(μ-2). (μ-n+1).δ.x^(δ-1) ]
=lim(x->+∞ ) x^(1-δ) .e^x/[μ(μ-1)(μ-2). (μ-n+1).δ ]
->+∞
=>
lim(x->+∞ ) e^x/x^μ ->+∞