Limits are well-defined by logical formula.
For example, if you want to express a function to tend to $l$ when $x \to \infty$, one will express as follow:
$$ \forall \epsilon>0, \exists A>0, (x>A\implies f(x)\in[l-\epsilon,l+\epsilon]) $$
What it says for that case, is sayint that $f$ tends to $l$ when $x$ is becoming bigger and bigger, means that if you take a small intervall around the limit $l$, you can find a number, above which $x$ is such that $f(x)$ is close to $l$ with a $\epsilon$ uncertainty.*So the notion of limit here stands with getting closer and closer to a value, and when working with infinity, talking about $\infty$ and $\lim$ without recalling the definition above becomes an habit.
Concerning $\pi$
$\pi$ is irrationnal, which means it has an infinite number of digit, and the sequence of its digit isn't periodic. So in practice, real concret world, we cannot "get" the infinite number of all digits of $\pi$ exactly (we can calculate each digit, but in our finite world, we do not have enough time to have all (infinite number) of digits of $\pi$). Yet the mathematical object of $\pi$ being a real irrational number $(\in \mathbb{R} -\mathbb{Q}$ if you use those symbols for now) is perfectly define and is an exact number in the abstract world of math.
For example one definition of $\pi$ can be :
$$\pi \ \text{is such number that for any circle of radius} \ R > 0,\ \pi=\dfrac{\text{Perimeter}}{\text{Diameter}} $$
Conclusion and opening
Limits work so well because they are a mathematical creation. As you see in the first definition of a limit when $x$ goes to $+\infty$, there is no symbol $+\infty$. In other words, here, we define a limit as a process to get closer and closer to a value, but in general, as you see in the definition above, we do not write $x=+\infty$. They are branch of mathematics where you can write it, but at this step it isn't relevant or correct in fact.
If you want to open your knowledge on this notion of exactitude, computability of a number, you can explore the difference between the following set of numbers, and ask yourself, which one of you can write all the digit of all the elements :
- Positive Integer $ \mathbb{N}$
- Integers numbers $ \mathbb{Z}$
- Decimal numbers $ \mathbb{D}$
- Rational numbers $ \mathbb{Q}$
- Real numbers $ \mathbb{R}$
- Irrational numbers $ \mathbb{R}-\mathbb{Q}$
At last if you feel very comfortable, with all the current infinity notions, and want to focus on even more subtle notion of "is a number that we know existing reachable in practice (theoretically or practically)" (only if you feel already well with the answers to your current questions), then look at the computable number definition and it's link with Turing's machine. https://en.wikipedia.org/wiki/Computable_number
