When you start playing with infinities phrases like "indistinguishable from 0" are not the same as "equal to 0." In this case the probability is being handled as a limit. One could consider the probability of 7 being picked is $1/\aleph_0$, however, $\aleph_0$ is what is called a "limit cardinal," and infinitesimals generated from limit cardinals are not included in the set of real numbers, so we have to be creative.
Consider a series defined by $1/x$ as we vary x:
So what's $1/\aleph_0$? It can't be zero, because that doesn't satisfy the rules of arithmetic $0 * \aleph_0 = 0$, but basic reorganization of the algebra shows that $1/x * x = 1$ for all x ($x \neq 0$).
The concept of a limit is used to rigorously explore this intuitive understanding that there is some limiting thing out there for $1/x$ that equals 0. In very intuitive and imprecise wording, this limit is the number "approached" by a series, but typically never actually achieves it.
Thus, when given an infinite number line and asked to pick a random value from it, the probability of any given number being picked is very very small. In fact, so small that the real numbers can't even describe how small it is. To capture this concept of smallness, we often look at the limit of the probability of a number being picked as we grow the set towards a cardinality of $\aleph_0$. This limit is zero. This is not to say the probability of picking any given number is zero, just that any process used to describe the meaning of picking randomly from an infinite set that starts from smaller sets and builds larger ones from there (typically using induction) will describe a series of probabilities that get closer and closer to 0 as the sets get larger.
The more formal definition of this limit could be phrased with some of the calculus you mention, using the epsilon-delta definition of a limit. By this definition, if you pick any arbitrarily small "epsilon," and try to determine which sized sets result in a probability of picking 7 below this epsilon, I can identify some set size for which all larger sets must yield probabilities smaller than this epsilon bound.
Of course, you probably don't need the formal definition much. Sometimes intuition is enough, but its good to know there's a formal definition coming later down the line for you!