Question

求极限！需过程

Answer 1

x->0
cos(x^2) ~ 1 -(1/2)x^4

lim(x->0) [cos(x^2)]^ (3/x^4)
=lim(x->0) [ 1 -(1/2)x^4 ]^ (3/x^4)

let
y = (1/2)x^4

lim(x->0) [cos(x^2)]^ (3/x^4)
=lim(x->0) [ 1 -(1/2)x^4 ]^ (3/x^4)
=lim(y->0) (1 -y)^ (3/(2y) )
=e^(-3/2)

追问

追答

By Taylor's expansion
f(x) = f(0) +[f'(0)/1!]x+ [f''(0)/2!]x^2 +......

x->0
sinx ~ x
(sinx)^2 ~x^2

lim(x->0) e^( -3(sinx)^2/x^4)
=lim(x->0) e^( -3/x^2 )
=0