Question

设当x→0时，a(x)＝kx²为β(x)＝√1+xsinx－√cosx是等价无穷小，则k＝?

Answer 1

x->0

sinx =x+o(x)

√(1+xsinx) = √(1+x^2+o(x^2)) = 1+(1/2)x^2 +o(x^2)

cosx = 1-(1/2)x^2 +o(x^2)

√cosx =√[1-(1/2)x^2 +o(x^2)] = 1- (1/4)x^2 +o(x^2)

√(1+xsinx) -√cosx =(3/4)x^2 +o(x^2)

α(x)＝kx^2 为 β(x)＝√(1+xsinx)－√cosx 是等价无穷小

=>

k=3/4