limx趋向于无穷(cos1/x)^(x)的极限
问题描述：
limx趋向于无穷(cos1/x)^(x)的极限
答
应该是
lim(x→0)[cos(1/x)]^x,
先计算
lim(x→0)x*ln[cos(1/x)]
= lim(t→inf.)(1/t)*ln(cost) (令 t=1/x)
= lim(t→inf.)(1/t)*(cost-1) (等价无穷小替换)
= ……
= 0,
因此
lim(x→0)[cos(1/x)]^x
= e^lim(x→0)x*ln[cos(1/x)]
= e^0
= 1.