Suppose $X$ is open in $\mathbb{R}^m$, $F$ is a Banach space and $f:\;X\to F$ is $m$-times differentiable. Then show $$ \partial^q f(x)\in\mathcal{L}^q_{\text{sym}}(X,F) \qquad\text{for}\;x\in X\;\text{and}\;1\leq q\leq m. $$
The exercise gives a hint to prove the case $q=m=2$. It says it suffices to prove that $$ \underset{\substack{s\to0\\s>0}}{\lim}\frac{f(x+sh_1+sh_2)-f(x+sh_1)-f(x+sh_2)+f(x)}{s^2}=\partial^2f(x)[h_1,h_2]. $$ By this hint, I obtain \begin{align} &\underset{\substack{s\to0\\s>0}}{\lim}\frac{f(x+sh_1+sh_2)-f(x+sh_1)-f(x+sh_2)+f(x)}{s^2}\\ &=\underset{\substack{s\to0\\s>0}}{\lim}\frac{1}{s^2}\left[\partial f(x+sh_1)[sh_2]+o(||sh_2||)-\partial f(x)[sh_2]-o(||sh_2||) \right]\\ &=\underset{\substack{s\to0\\s>0}}{\lim}\frac{1}{s^2}\left[\partial^2f(x)[sh_2,sh_1]+o(||sh_1||)-o(||sh_2||)+o(||sh_2||)\right] \end{align}
How to prove this equals $\partial^2f(x)[h_2,h_1]$.