已知f(x)在x=0处连续,则a＝不太会算cosx^(1/x^2)的极限,

Question

已知f(x)在x=0处连续,则a＝
不太会算cosx^(1/x^2)的极限,

Answer 1

lim(x->0) (cosx)^(1/x^2)
=lim(x->0) [1+(cosx-1)]^[ 1/(cosx-1) * (cosx-1)/x^2 ]
=lim(x->0) {[1+(cosx-1)]^[1/(cosx-1)]}^ [(cosx-1)/x^2 ]
∵lim(x->0) {[1+(cosx-1)]^[1/(cosx-1)]} = e
lim(x->0) [(cosx-1)/x^2]=lim(x->0) [(-x^2/2)/x^2]= -1/2
=e^(-1/2)
∵ 在x=0处连续,则：lim(x->0) (cosx)^(1/x^2) = f(0)
∴ a = e^(-1/2)