Taylor's formula for the problem of limit SiNx ^ (- 2) * x ^ (- 2) when x approaches 0, we use Taylor's formula of order 3 How do you calculate it?

Taylor's formula for the problem of limit SiNx ^ (- 2) * x ^ (- 2) when x approaches 0, we use Taylor's formula of order 3 How do you calculate it?

Taylor
sinx=x-x3/(3!)+x5/(5!)-x7/(7!)+……
It's unfolding
sinx^(-2)*x^(-2)
=x^(-4)-x^(-8)/(3!)+x^(-12)/(5!)

1. If f (x) is second-order differentiable, then why is f '' (ξ) in Taylor formula a function of X
Example: ∫ f '' (ξ) DX ≠ f '' (ξ) ∫ DX
2. Is the following correct? If not, please correct
(1) ξ in Rolle's theorem is a constant
(2) In Taylor's theorem, ξ is a function of X
(3) Taylor's formula with Laplace's remainder is used to prove, and Taylor's formula with pitot's remainder is used to calculate limit

I'm not impressed, but the first problem is: ξ will change with the change of X, of course, it's a function of X, so if there are two ξ: ξ 1 and ξ 2 in the title, don't think they are equal. They are all functions of X

It is proved that the equivalent infinitesimal of arctangent function is X

Proof: (the following are the limits under X - > 0, the input is inconvenient, omitted)
lim(arctanx/x)
=lim[1/(1+x^2)]
=1
Therefore, the equivalent infinitesimal of arctangent function is X