求极限limn→∞（cosx/n+sinx/n）^n，x≠0详细过程

Question

如题所述

Answer 1

做一道考研题吧
1^∞型
lim《n→∞》[cos(x/n)+sin(x/n)]^n
=lim《n→∞》[1+cos(x/n)+sin(x/n)-1]^n
=lim《n→∞》e^{n*[cos(x/n)+sin(x/n)-1]}
=lim《n→∞》e^{n*[1-(x/n)^2/2+(x/n)+o(1/n^2)-1]}
=lim《n→∞》e^{n*[(x/n)+o(1/n)]}
=lim《n→∞》e^{[x+o(1)]}
=e^x