Does the presence of irrational numbers pose any problems for the concepts of limits and continuity? Could someone discuss in an intuitive (not too formal) way whether irrational numbers like $\pi$ would pose any problems to the ideas of limits and continuity?
I'm not sure if they do, or not, but it does seem like it might be possible. 
 A: *

*They pose no problems. You can study question of limits and continuity on the rational numbers rather than the reals (although the results are somewhat different).

*On the other hand, the very notion of irrational numbers arose in part from the study of something like limits, so they're at least related. 
More details: 
The definition of continuity (or limits) is cleverly written: it says that $f$ is continuous at $a$ if for every number $e > 0$, there's a number $d$ such that for every $x$ with  $|x - a| < d$, we also have $|f(x) - f(a) | < e$. Informally: no matter how close ($e$) you say you want $f(x)$ to be to $f(a)$k, I can find a distance $d$ so small that if $x$ is at least that close to $a$, then $f(x)$ will be close enough to $f(a)$. You don't have to understand every word of that to see that nowhere in it is there any mention of rationals or irrationals. In fact, in choosing the numbers $d$ and $e$, we can let them range of all real numbers, or over just the rationasl, and the results well be the same. 
On the other hand, look at the sequence
3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...

of numbers $u_i$, where $u_i$ has the property that $10^i u_i < 10^i \pi < 10^i u_i + 1$,
and $10^i u_i$ is an integer. In other words, $u_i$ has "the first $i$ digits of the decimal expansion of $\pi$". 
In the rational numbers, this sequence does not have a limit (just as in the interval 
$0 < x < 1$, the numbers $1, 1/2, 1/3, \ldots$ have no limit). In the reals, there is a limit. In fact, for the reals, 
any increasing sequence of numbers that are all less than some fixed number $M$ have a limit, $L$, that is no greater than $M$.
Applying this to the $u_i$ sequence above, you can see that the $u_i$ are an increasing sequence, an that they are all less than, say, 10. That means that they have to have a limit. This property turns out to be really useful in proving theorems, which means that there are many theorems about the real numbers that are harder to prove for the rationals, or not true at all. 
As an example: if $a, b, c$ are real with $b^2 - 4ac > 0$, then the equation $ax^2 + bx + c = 0$ has solutions. 
That's not true for the rationals, as the case $a = 1, b = 0, c = 2$ shows. 
A: To the contrary, irrational numbers basically complete the concepts of limits and continuity for rational numbers.
If you take the rational numbers on their own, they are a great many "gaps" between them where the irrational numbers go. (This is rather hard to visualise, because it is both true that there are infinitely many other rational numbers between any two rational numbers and true that you have these gaps.) And one of the ways you can tell these gaps are there are via trying to take limits, because if you just look at the rational numbers there are many sequences that look like they "ought" to converge but don't. (The formal concept for what I'm calling "ought to converge" is Cauchy convergence - roughly speaking, a sequence is Cauchy when terms in the sequence get arbitrarily close together.)
For example: the sequence $3, 3.1, 3.14, 3.141, 3.1415,...$ is one composed solely of rational numbers, but it doesn't have a limit in the rationals (since that limit would be $\pi$). So by working with the notions of limits and convergence you can use the rational numbers to identify the spot in them where $\pi$ "ought" to be and isn't. And you can actually do this for any irrational number.
So in fact, one of the ways of defining real numbers is by saying "okay, let's extend the rational numbers so that every Cauchy sequence has a limit" (with some compatibility regarding when two Cauchy sequences have the same limit.) That means you add precisely the irrationals. As a result it's actually easier to work with limits and convergence in the reals than the rationals in many ways because unlike in the rationals, every sequence that "looks like it should have a limit" actually does.
