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

1

  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.