I'm trying to evaluate the following limit:
$\lim_{x\to 0}\frac{\sin\left(\int_{x^3}^{x^2}\Bigg(\int_0^t g(s^2) \, ds\right) \, dt\Bigg)}{x^8}$
for $g:[-1,1]\to\mathbb{R}$ differentiable function such that $g(0)=0$, $g'(0)=1$.
I developed $g$'s taylor expansion near $0$ and found out that $g(x)=x+o(x)$ where $o(x)$ is a function such that $o(x)/x \to_{x\to 0}0$.
Now, intuitively $o'(x^n)\sim o(x^{n-1})$ and $\int o(x^n) \, dx\sim o(x^{n+1})$ but how to formally justify it? I feel that I don't understand Taylor's theorem good enough.
By the way, I calculated (using my unjustified intuitions) that the limit above exists and equals to $\frac{1}{12}$.
Thanks for your help!