Truncated Taylor expansion example

Question

Let $f$ and $g$ be two functions infinitely continuously differentiable that have the following Taylor expansions:

$f(h)=h+o(h^3)$ and $g(h)=h+o(h^3)$. Then $(f-g)(h)=o(h^3)$ around $h=0$

Why is this statement false? I have no idea how to approach this problem...

EDIT: Apparently this statement is in fact true

Answer 1

This seems true to me:

$f(h) = h + o(h^3) \iff \lim_{h \to 0} \frac{f(h)-h}{h^3}=0$

So,

$\frac{f(h)-g(h)}{h^3}=\frac{f(h)-h}{h^3}+\frac{h-g(h)}{h^3}$

is a sum of two functions which converge to $0$ when $h \to 0$.

So, it's true that: $f(h)-g(h)=o(h^3)$