$\fbox{Hypothesis}$
EDIT: Let $\{t_n\}$ be a sequence of reals s.t. $t_n \rightarrow t_0$ with $t_n \ne t_0$.
Let $\{g_{t_n}\}$ be a sequence of integrable functions such that for $n$ large enough, $g_{t_n} \le g$ for some integrable $g$.
Then for a fixed function $g_{t_0}$, consider now the derived sequence of functions:
$\left\{ \frac{(g_{t_n} - g_{t_0})}{t_n - t_0} \right\} = \{d_n\}$
$\fbox{Problem}$
How do I show that $\{d_n\}$ is also dominated for $n$ large enough by some other function (no doubt derived in some way from $h$)?