Suppose we have a sequence of functions $\{ f_n \}$: $\mathbb{R}^p \to \mathbb{R}^q$ differentiable at $x$.
Let $x_n \to x$, and consider the Taylor expansion
$f_n(x_n) = f_n(x) + \nabla f_n(x)(x_n-x) + R_n$
Under which conditions do we have $R_n = o(|x_n - x|)$ (Peano form for the remainder)?
Additionally, if the second derivative is defined in a neighborhood of $x$, under which conditions do we have $R_n = O(|x_n - x|^2)$ (Mean-value form for the remainder) ?
My intuition is that the remainder form is correct if the family $\{\nabla f_n\}$ is equicontinuous at $x$, but I'm not sure how I would prove this.
For the first part, what is needed is "equidifferentiability", as defined in
Moore, R.H., 1966. Differentiability and convergence for compact nonlinear operators. Journal of Mathematical Analysis and Applications, 16(1), pp.65-72.
With equidifferentiability at a point $x_0$, you have $\max_n(|R_n|) = o(|x_n-x_0|)$, which let you do the Taylor expansion required.
Alternatively, we can have equicountinuity of $\{ \nabla f_n(x)\}$ in a $r$-neighborhood of $x_0$. Proof here.
For the second part, if the second derivative $\{ \nabla^2 f_n(x)\}$ exists in a neighborhood $U_x$ of $x_0$ and $max_{x \in U_x}\nabla^2 f_n(x) = O(1)$ as $n \to \infty$, then we have $R_n = O(|x_n-x_0|^2)$ from the mean-value theorem applied to each $q$ component of $f_n$ separately.
Again, this can be strengthened to $R_n = (x_n - x_0)^T\nabla^2 f_n(x_0)(x_n - x_0) + o(|x-x_n|^2)$ if we assume equidifferentiability of the second derivative at a point $x_0$.