令原式=y
则lny=4ln(cosx)/x^2
x→0,ln(1+x)和x是等价无穷小
所以ln(cosx)~cosx-1
而1-cosx和x^2/2是等价无穷小
所以cosx-1~-x^2/2
所以lim(x→0)lny=lim(x→0)4(-x^2/2)/x^2=-2
所以lim(x→0)y=1/e^2