Using Dominance Convergence Theorem in derivation of Leibniz's rule

Question

I really need help understanding how Dominance Convergence Theorem applies to the following limit:

$$ \underset{\Delta t\rightarrow 0}{lim}\int_{a(t)}^{b(t)}\frac{f(x, t+\Delta t)-f(x,t)}{\Delta t}dx $$

So that we can change the limit to the following integral:

$$ \int_{a(t)}^{b(t)}\underset{\Delta t\rightarrow 0}{lim}\left ( \frac{f(x, t+\Delta t)-f(x,t)}{\Delta t} \right )dx $$

For clarification as to where I got this limit from, I was watching a video on the derivation of Leibniz's rule (link) and this was the only portion that I was unable to understand. The person doing the derivation just said we can switch the integral and limit, but I would like to understand why. Looking into it on my own, I understand that Dominance Convergence Theorem is used to switch a limit and integral, I just don't understand why this theorem can be applied in this specific case.

Answer 1

Fix $t$. Let $g_h(x) = \frac{f(x, t+h) - f(x,t)}{h}$. If there exists a function $g^*$ that

• is integrable on the interval $[a(t), b(t)]$, and
• is such that $|g_h(x)| \le g^*(x)$ for all $x \in [a(t), b(t)]$,

Then $\lim_{h \to 0} \int_{a(t)}^{b(t)} g_h(x) \, dx = \int_{a(t)}^{b(t)} \lim_{h \to 0} g_h(x) \, dx$ (i.e. you can switch the order of limit and integration.)

I didn't watch the video, but I assume there are some assumptions on $f$ such that the above conditions hold.