$\lim_t \int (F(x t) - F(0)) / xt dx$ and dominated convergence

First of all, for all $x\in[0,1]$ we have $$ \lim\limits_{t\rightarrow 0}\frac{F(xt)-F(0)}{xt}=F'(0) $$ Suppose without loss of generality that $t\leqslant 1$ and let $\delta>0$ such that $\left|\frac{F(u)-F(0)}{u}-F'(0)\right|\leqslant 1$ for all $u\leqslant\delta$ (such a $\delta$ exists because $F$ is differentiable at $0$). Then $\left|\frac{F(xt)-F(0)}{xt}-F'(0)\right|\leqslant 1$ when $xt\leqslant\delta$, that is when $x\leqslant\frac{\delta}{t}$. On the other hand, if $x>\frac{\delta}{t}$, then $\left|\frac{F(xt)-F(0)}{xt}\right|\leqslant\frac{2\|F\|_{\infty,[0,1]}}{\delta}$ because $0\leqslant xt\leqslant 1$. Therefore, $$ \left|\frac{F(xt)-F(0)}{xt}\right|\leqslant |F'(0)|+1+\frac{2\|F\|_{\infty,[0,1]}}{\delta} $$ for all $x\in[0,1]$ and $t\in[0,1]$. You have your dominating function, thus you can apply the dominated convergence which gives you the limit : $$ \lim\limits_{t\rightarrow 0}\int_0^1 \frac{F(xt)-F(0)}{xt}dx=F'(0) $$