Limit of a function with a defined integral I have the next limit:
\begin{equation}
\lim_{x\to\ 0}\displaystyle\frac{\displaystyle\int_0^{x^2}{\frac{1-\cos{t^2}+at^4}{t}}dt}{(1-\cos{(\frac{x}{3})})^4}
\end{equation}
I've tried to solve it by doing L'Hôpital, and then I had the variable $x$ everywhere, so I applied some infinitessimals equivalences and then I realised that the limit is $0$, and the value of $a$ doesn't matter (because when I simplified everywhere, the $x$ in the numerator has degree 8, and in the denominator has degree 7, so it went to $0$).
What I have say above, it's correct or I must obtain a different solution? Thanks!

Proceed with L'hopital rule...: Differentiate both numerator and denominator and simplify to get:
$2x\cdot \dfrac{1-cosx^4 + ax^8}{x^2}\cdot \dfrac{1}{4(1-cos(\frac{x}{3}))^3\cdot \frac{1}{3}\cdot sin(\frac{x}{3})} = L$. Using: $sinx \sim_0 x$, $1 - cosx = 2sin^2(\frac{x}{2})\sim_0 2\cdot \left(\frac{x}{2}\right)^2$, and the substitution of $ y = x^4$, we get:
$L \sim_0 \dfrac{9\cdot 6^6}{16}\cdot \left(\dfrac{1 - cosy}{y^2} + \dfrac{ay^2}{y^2}\right) \to \dfrac{9\cdot 6^6}{16}\cdot \left(\dfrac{1}{2} + a\right)$
