[Math] When can the order of limit and integral be exchanged

Question

1. I was wondering for a real-valued function with two real variables, if there are some theorems/conclusions that can be used to decide the exchangeability of the order of
   taking limit wrt one variable and taking integral (Riemann integral,
   or even more generally Lebesgue integral ) wrt another variable,
   like $$\lim_{y\rightarrow a} \int_A f(x,y) \, dx = \int_A
   \lim_{y\rightarrow a} f(x,y) \,dx \text{ ?}$$
2. If $y$ approaches $a$ as a countable sequence $\{y_n, n\in
   \mathbb{N}\}$, is the order exchangeable when $f(x,y_n), n \in \mathbb{N}$ is uniformly convergent in some subset for $x$ and $y$?

3. How shall one tell if the limit and integral can be exchanged in the following examples? If not, how would you compute the values of the integrals:

   • $$\lim_{y\rightarrow 3} \int_1^2 x^y \, dx$$
   • $$ \lim_{y\rightarrow \infty} \int_1^2 \frac{e^{-xy}}{x} \, dx$$

Thanks and regards!

Answer 1

The most useful results are the Lebesgue dominated convergence and monotone convergence theorems.