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.