函数f(x)=(cosx)^(1/x^2)(x不等于0),f(x)=a(x=0时),在x=0处连续,求a.

Question

函数f(x)=(cosx)^(1/x^2)(x不等于0),f(x)=a(x=0时),在x=0处连续,求a.

Answer 1

答：
x≠0,f(x)=(cosx)^(1/x^2)
x=0,f(x)=a
f(x)在x=0处连续,则有：
lim(x→0) (cosx)^(1/x^2)=f(0)=a
lim(x→0) [1-2sin²(x/2)]^(1/x²)=a
lim(x→0) [1-2*(x/2)²]^(1/x²)=a
lim(x→0) [(1-x²/2)^(-2/x²) ]^(-1/2)=a
所以：e^(-1/2)=a
所以：a=1/√e