Which theorem or rule is this:
$\displaystyle \int_{b}^{c} \lim_{x\to a} f(x) dx = \lim_{x\to a} \int_{b}^{c} f(x) dx$?
When you can a limit which is inside a integral add before an integral.
Let's consider
$\displaystyle \int_0^2 \lim_{x \to 0}x^2dx$ and $\displaystyle \lim_{x \to 0} \int_0^2 x^2dx$
The LHS and RHS are clearly not equal. I'd say it is rare when the two would be equal. On the LHS, you are taking the integral of a constant. On the RHS, you are taking the limit of a constant.
Do you mean something like this:
$\displaystyle \lim_{n \to \infty} \int_a^b f_n(x)dx = \int_a^b \lim_{n \to \infty} f_n(x)dx$?
That is a very different equation from what you wrote. What you wrote, the limits turn the function into a constant. In what I wrote, the limit turns a family of functions into a limit function. Check the Lebesgue Dominated Convergence Theorem and the Monotone Convergence Theorem to start.
Note that the left hand side is the integral of a constant, while the right hand side is the limit of a constant.
Moreover, on the right hand side the $\displaystyle x$ in $\displaystyle \lim_{x \to a}$ is not the same $\displaystyle x$ as the variable of integration $\displaystyle \int_b^c f(x)\,\mathrm dx$
Unfortunately, this is not the correct medium for a lecture in real analysis. I'm sure there are plenty of texts, YouTube videos, etc. that could do the topic of convergent sequences (or families) of functions more justice than I could.
A quick google search found this:
http://www.personal.psu.edu/auw4/M401-notes1.pdf
Oh, I think I got it now.
So for example we have a set A, which contains a sequence of functions or should I say it's a sequence of functions:
$\displaystyle A = \{f_n, f_m, f_o\}$
where lets say:
$\displaystyle f_n = nx$
$\displaystyle f_m = cos(xm)$
$\displaystyle f_o = 3o^x$
In those examples we have just one function in the set.
I don't understand what you wrote at all. You have a set of three functions, $f_n, f_m, f_o$. That is not a sequence of functions (a sequence is a countably infinite set where there is a progression. There is an initial function, and then each function knows the function that comes after it. Example: $f_0, f_1, f_2, \ldots$ is a sequence of functions.
To clarify this example further, suppose:
$f_0(x) = 0x = 0$
$f_1(x) = 1x = x$
$f_2(x) = 2x$
$f_3(x) = 3x$
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$f_n(x) = nx$
That is a sequence of functions.
There are an infinite number of functions.
If we fix a point, say $x=c$ for some $c \in \mathbb{R}$, then we look to see does the sequence of numbers, $f_0(c), f_1(c), f_2(c), \ldots$ converge to a single point? It only converges for the point $x=0$. Otherwise, it does not converge.
Alternately, if we look at the functions:
$g_1(x) = \dfrac{x}{1}$
$g_2(x) = \dfrac{x}{2}$
$g_3(x) = \dfrac{x}{3}$
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$g_n(x) = \dfrac{x}{n}$
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Consider any constant $c \in \mathbb{R}$. We have:
$\displaystyle \lim_{n \to \infty} g_n(c) = \lim_{n \to \infty} \dfrac{c}{n} = c\lim_{n \to \infty} \dfrac{1}{n} = c(0) = 0$
This converges point-wise to the zero function. So, we can say: $\displaystyle \lim_{n \to \infty} g_n(x) = g(x) = 0$.
Consider the sequence of functions:
$h_1(x) = x+\dfrac{1}{1}$
$h_2(x) = x+\dfrac{1}{2}$
$h_3(x) = x+\dfrac{1}{3}$
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$h_n(x) = x + \dfrac{1}{n}$
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This sequence of function converges to the function $h(x) = x$.