On
The infinity in calculus is not a cardinal. It does not correspond to a size of a set. Much like $\pi$ does not correspond to any cardinality of a set.
The "infinity of the real numbers" and "the infinity of the integers" are infinite cardinals which represent sizes of sets. There are more, many many more infinite cardinals which represent sizes of other sets. But those cardinals are not real numbers, and not the limit of real numbers.
The infinity you see in calculus represents a potential infinity, which is the idea that something is growing indefinitely. If you want to think about the real numbers as a timeline, by taking the limit of something towards $\infty$ we essentially ask whether or not it stabilizes over time. But the limit itself is the limit of finite, real numbers. Not of cardinals, not of anything else.
That been said, there are several notions of infinity that you can find in calculus. From unsigned $\infty$, to signed $\pm\infty$, to other types of infinity that you might meet.
Your friend sounds like he is misguided in two different ways.
The first thing to note is that the notion of "infinite cardinal number" -- e.g. the numbers we use to quantify the size of the set of integers ($\aleph_0$)or the set of real numbers (sometimes we call it $2^{\aleph_0}$, other times $\mathfrak{c}$, or we might just write $|\mathbb{R}|$)-- is a very different kind of number than the notion of a "point at infinity".
In particular, $\aleph_0$ and $\mathfrak{c}$ have absolutely nothing to do with the numbers $+\infty$ and $-\infty$ that you see in calculus. $+\infty$ and $-\infty$ are more geometric notions, and can be intuitively viewed as if you added two endpoints to the interval of all real numbers, and this idea has nothing to do with the sizes of sets.
The second thing to note is that there aren't just two infinite cardinal numbers: the cardinality of the set of all real-valued functions, for example, is a cardinal number that is even greater than $\mathfrak{c}$. And even that infinite number is tiny compared to how large cardinal numbers can get.
However, your friend might have meant something different entirely; he could have been thinking more about the ideas of a "discrete set" and "continuum". He may have been thinking specifically about the difference of how a sequence approaches a limit, and how a function approaches a limit. While both are instances of the same idea (a limit), there are qualitative differences in how they behave, although I'm having trouble thinking of an example to demonstrate.
(but even then, these two cases aren't exhaustive; they're just the two cases you would run into most often)