# The Idea of a Limit

An example of a sequence of numbers is the following: $1,4,9,16,25,\dots$ or $5,1,7,2,6,4,2,\cdots$. As you can see, one sequence is quite ordered in that it has a certain rhythmical structure to it, while the other appears to be seemingly random. We usually prefer to study those sequences with a type of structure to it, and we do this by simplifying the way we write these types of sequences.

The sequence $(n^2)_{n\in \{1,2,3,4,\dots \} }$ is an alternative way of writing $1,4,9,16,\dots$. In other words, we are replacing $n$ by consecutive numbers i.e. $(1)^2,(2)^2,(3)^2,(4)^2,\dots$.
For the more mathematically literate, a sequence is simply an injective function $f$ from $\mathbb{N}$ explicitly defined by $f(n)=n^2$ for $n \in \mathbb{N}$

Here are a couple of examples of sequences which might be interesting:

1. $(\frac{1}{n})_{n \in \mathbb{N}} \equiv 1,\frac{1}{2},\frac{1}{3},\dots$
2. $(2^n)_{n \in \mathbb{N}} \equiv 2^1,2^2,2^3,\dots$
3. $(\frac{n^2+1}{n})_{n \in \mathbb{N}} \equiv \frac{1^2+1}{1},\frac{2^2+1}{2},\frac{3^2+1}{3},\dots$

The next thing to come to grasps with is the idea of a LIMIT of a sequence.
A limit is a particular value that the sequence approaches but does not necessarily reach. Let’s consider Example 1, if we manually compute the first few values of this sequence, we obtain that:

$(\frac{1}{n})_{n \in \mathbb{N}} \equiv 1,0.5,0.33\bar 3,0.25,0.2,\dots$ has terms which appear to get smaller and smaller the farther along the sequence we go

Does it ever go negative? Convince yourself why this can’t ever be so.

So what do we know? We know that we have a sequence of positive numbers that are getting progressively smaller and smaller, and that they never end up becoming negative. What is the intuitive answer to what this sequence is ‘tending’ to?

ZERO!

We can represent this concept more formally by saying that $lim_{n\to \infty} \frac{1}{n} = 0$ or in other words, as $n$ gets larger and larger, the sequence $\frac{1}{n}$ tends to 0.

This appears to be quite wishy washy for some, especially Mathematicians and this was in fact was quite a hotly debated area that desperately required some form of mathematical rigour, as it was essential in giving justification to the rest of mathematical analysis. The following preferred method of rigorously defining the ‘limit’ is the $\varepsilon - \delta$ (Epsilon-Delta) definition.  It goes as follows:

$lim_{n\to \infty} a_n = a \iff \forall \varepsilon>0, \exists N \in \mathbb{N}, s.t. (a_n - a)<\varepsilon, \forall n > N$

Difficult to digest? Here it is in English:

“A sequence $a_n$ is defined to tend to a limit $a$, if from any distance from the limit that you pick, I am able to find a point in the sequence, that will remain as close to the limit as the distance you gave me, or closer, for the rest of the sequence.”

What does this mean? Well, pick any distance that you want, let’s say a distance of $0.0001$. I can be certain that after 10,000 steps in the sequence, it will remain as close to $0$ as $0.0001$ or closer

$\cdots,\underbrace{0.0001}_\frac{1}{10000},\underbrace{0.000099}_\frac{1}{10001},\dots$

This works no matter what distance you give me, which means that $0$ is the limit of the sequence.

Why isn’t $-1$ the limit of this sequence for example? Well, if you give me a distance of 1.5, then I’d be able to find you a point past which the sequence remains as close to $-1$ as that distance.

But the moment you give a distance that is smaller than 1, I cannot. If you tell me 0.5 for example, no where in the sequence will it ever be possible to get as close to the number $-1$ as 0.5 or closer because it never goes past 0.

Since it doesn’t work for ALL distances, then it is not its limit!

Below I have linked a really great video by 3Blue1Brown, an innovator in the way Mathematical ideas are presented to the public sphere, where he explains how Math is more an intuitive and creative subject than meets the eye 🙂 He also explains sequences, which is relevant to what we were talking about today.

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