The Aleph numbers and infinity in calculus.
Mathematical objects that are called "numbers" are usually representatives for some amount, and we want to say that it is the amount that matters.
For example, if I cut a meter long stick into two equal pieces, each of them will be exactly as long exactly as putting five $10$ cm rulers one after another. It is not the object but rather the length which plays along.
The real line's infinity: Nonnegative real numbers can be thought as abstract notion of length, while negative numbers can be thought of as either "making things shorter" or as formal objects to allow additive inverses.
The infinity on the real line represents an abstract notion of "being longer than any other length". You can think of it formally as being larger than any finite length: something has "infinite" length if it is longer than an object of length $1$, an object of length $2$, and so on.
It is important to remember that this is just a formal symbol, that $\infty$ is added to $\mathbb R$ to say "If you reached this point - you've gone too far".
When we say: $$\lim_{n\to\infty}\frac{x^n}{e^n} = 0$$
It may contradict my previous statement, since those are both "infinite" numbers. However in this limit we compute the behavior of the ratios when taking bigger and bigger "lengths".
The story of $\aleph$: On the other hand we have cardinal numbers. These are numbers which measure the most primitive notion of size, namely "how many oranges are in this pile?". In finite numbers the notions coincide, $10$ oranges can be used to measure a $10$ foot distance. However when talking about an infinite number of oranges the rules of the game change - as they usually do with infinite objects.
The idea behind the aleph numbers is to measure how many oranges are in some pile, or how many cats are in the bag. Now it does not matter if your bag says "Natural numbers" or "Integers" or any other countable set. It means just that there are $\aleph_0$ many elements in the set.
Of course, $\aleph_0$ is the size of the set which has more elements than any finite set. Much like the $\infty$ idea is the length beyond all finite lengths. However the $\aleph$ numbers have a very accurate definition, they do not rely on some limit but rather on a precise definition. Their existence is deduced from the axioms which assert their existence, unlike $\pm\infty$ which is often added to $\mathbb R$ as formal symbols to signify "the end of the line".
This makes the two notions of infinity very different. If we could compare $\infty$ and $\aleph_0$ and $2^{\aleph_0}$ (the latter being the cardinality of the continuum) we can notice several things:
To "reach" $\infty$ we need only to take countably many steps, that is $\lim\limits_{n\to\infty}\ n$; however we pass over $2^{\aleph_0}$ many elements.
On the other hand, to reach the size of $\aleph_0$ we cannot pass over more than countably many elements. If we skip over uncountably many elements then we have uncountably many elements, that is more than just $\aleph_0$ many of them.
In contrast, again, even if we do take a countable collection of countable piles we cannot "reach" the continuum. In that sense, in order to meet the continuum (while increasing cardinality because this is what we measure) we have to take uncountably many steps, this is regardless to its actual value as a cardinal number - it can never be the countable union of strictly smaller cardinalities.
To conclude, there are several notions of "number" in mathematics, they all have about one thing in common: they give us some "measurements" on mathematical objects; however some of them behave very differently from one another. There is no need to expect there will be a single notion of infinity either, and there is not just one notion. There are many. However all infinities have one thing in common: they are always "larger than any finite measurement".
Some reading material on this site:
- Intuition about the size of $\aleph_k$ with $k>1$
- Comparing infinite numbers
- Is infinity a number?
- Why is $\omega$ the smallest $\infty$?
The $\aleph_0$, $\aleph_1$, etc are cardinal numbers, and are specific mathematical objects. The notation $\infty$ used in calculus is more of a notation, and usually does not refer to a particular mathematical object. For example, $[6, \infty)$ is $\{ x : x \in \mathbb R, x \geq 6\}$. The $\infty$ by itself doesn't denote anything, just like the $[$ by itself doesn't denote anything. When one says $$\lim_{x \downarrow 0} \frac 1 x = \infty$$ the $\infty$ does not refer to any object, but the whole thing means that one can make $1/x$ arbitrarily large by making $x$ small (formally, for all $N \in \mathbb R$ there exists $\epsilon > 0$ such that for all $x \in (0,\epsilon)$, $1/x > N$).
Note that sometimes one might use $\infty$ as a convenient name for something (instead of something less suggestive like $x_0$ or some such), for example as the element one adds to the complex plane to get the Riemann sphere.
They aren't really directly comparable; they have fundamentally different motivations.
The "infinity" of algebra and calculus is motivated by the idea of considering larger and larger numbers, and deciding what happens "in the limit". It is often insisted that infinity is not a number, and for a variety of good reasons; arithmetic just usually isn't well-defined when you add infinity to the mix. When infinity is defined as a number, in most conceptions (for real numbers, at least), there is either only one or two infinities (sometimes people will merge $\pm\infty$), or $\infty + 1 \neq \infty$, and there are many infinities of the some order of magnitude, as well as $\infty^2$, and so on, numbers of vastly larger magnitude. In the latter case, arithmetic is well-defined.
The cardinal numbers, by contrast, are just a toolkit for describing how large a set is. It does not describe a limit of smaller things; it describes a static collection's behavior with respect to the equivalence relation of one-to-one correspondence. Subtraction is not well-defined, but addition, multiplication and exponentiation are all well-defined. In this scheme, when $\alpha$ is an infinite cardinal, $\alpha + 1 = \alpha$, as we might intuitively expect. However, there is not only one infinite number; $2^\alpha$ is strictly larger than $\alpha$, according to Cantor's Theorem.
One more collection of infinite numbers related to set theory that might interest you are the ordinal numbers. I'm assuming you haven't seen them before, nor are you familiar with well-ordering.
Here's a thought experiment: imagine counting $0,1,2,3\cdots$. You would never reach it, but let's say you speed up the clock so that you finish up all natural numbers in a finite interval of time. Then the next number you'll count is $\omega$ (for the sake of our experiment). Note that you never said $\omega-1$, and in fact, no such number exists. But then you continue, $\omega+1, \omega+2, \cdots$. Speeding up the clock again, you reach $\omega \cdot 2$. In the same way, you get $\omega\cdot3$, $\omega\cdot4$, etc. Those numbers approach $\omega^2$, and then similarly you get $\omega^3, \omega^4, \cdots$, approaching $\omega^\omega$, and it keeps going. Now, all of the numbers I've described have countably many predecessors, and in order to be able to describe an ordinal number exactly in this manner, it would have to be countable. However, it turns out that if we define a collection of numbers in exactly this manner in set theory, where each number is characterized by the set of predecessors, there are, in fact, numbers with uncountably many predecessors, of any cardinality. These are the ordinal numbers. Have a look at the wikipedia articles on ordinals and well-ordering if you're interested.