设a,b∈R+,则lim(a^n+b^n)/(a+b)^n=

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设a,b∈R+,则lim(a^n+b^n)/(a+b)^n=

设a,b∈R+,则lim(a^n+b^n)/(a+b)^n=
设a,b∈R+,则lim(a^n+b^n)/(a+b)^n=

设a,b∈R+,则lim(a^n+b^n)/(a+b)^n=
(1)当a=b时,
原式=lim(n->∞)[2a^n/(2a)^n]
=lim(n->∞)[1/2^(n-1)]
=0;
(2)当alim(n->∞)[(a/b)^n]=0,lim(n->∞)[(a/b+1)^n]=∞
原式=lim(n->∞)[((a/b)^n+1)/(a/b+1)^n] (分子分母同除b)
=(0+1)/∞
=0;
(3)当a>b时,同理可得,原式=0;
故综合(1)(2)(3)知,当a,b∈R+时,则lim(n->∞)[(a^n+b^n)/(a+b)^n]=0.