Calculus (II): What Is A Limit? How To Really Grasp It?

Every beginning calculus student faces the following challenging question: What is a limit? The notion of the limit is arguably the second most important one to grasp after that of a function. In my first entry in the calculus series, I covered the ground fundamentals of a function.

This essay is the second entry in the calculus series, where I will be diving into a deep discussion about the notion of a limit. If you did not know or realise, the typical derivative from differential calculus is actually a limit. And so is the typical integral from integral calculus.

So, before even starting out with derivatives and integrals, it makes absolute sense to strengthen one’s fundamental understanding of the limit. Having said this, I mentioned “typical derivative” and “typical integral” for a reason.

You see, calculus exists in two prominent variants: the way schools teach calculus (the conventional way) and the Nonstandard variant. The latter does not use the notion of the limit whereas the former does.

In this essay, we will only be discussing conventional calculus (don’t worry, I’ll cover the Nonstandard variant in a fresh essay). Without any further ado, let us begin.

This essay is supported by Generatebg

Discrete Variables and Continuous Variables

To begin, we need to touch upon these two relevant variable types. This is because they help one grasp the notion of a limit quicker.

A discrete variable is one that jumps from one value to another. Consequently, it does not capture what lies between any two jumps. Examples of discrete variables would be the number of people in a group, the outcome when you roll a pair of dice, etc.

You won’t normally say that you have 2.7 members in your family or you rolled a pair of dice to get an outcome of 5.65. On the other hand, a continuous variable varies continuously by taking real number values.

Examples of continuous variables include temperature change over time, body weight change over time, etc. When the weight of a person changes from 60 Kilograms to 61 Kilograms, it goes through an infinite number of real number values as it shifts.

Therefore, discrete variables typically operate on finite sets whereas continuous variables operate on infinite sets. Having covered discrete and continuous variables, let us now shift our focus towards sequences and series.

Sequences, Series, and Sums

Although ‘limit’ is a calculus concept, we are at a point where it is unproductive to use calculus to understand the concept. Instead, we will be using the notion of series to understand it. Actually, many modern calculus textbooks take the same approach.

If you are not familiar with what a series is, do not worry; we will cover that in a moment. Let us start with a sequence. Any set of numbers laid out in some order is known as a sequence. For example, consider this sequence of the first five positive non-zero integers:

1, 2, 3, 4, 5

Since there are just 5 numbers in this sequence, this is known as a finite sequence. Sequences can also be infinite. An example of an infinite sequence would be that of ALL positive non-zero integers:

1, 2, 3, 4, 5,…

If the terms (numbers) of a sequence are summed up, it leads to a series. Consequently, we can have finite and infinite series:

x = 1 + 2 + 3 + 4 + 5

y = 1 + 2 + 3 + 4 + 5 + …

In the above example, the variable x equates to a finite series (whose sum is just 15 by the way) and the variable y equates to an infinite series. It is quite clear how one can sum the terms of finite series.

However, how does one sum up the terms of an infinite series? You see, that question gets us one step closer to our goal of understanding the notion of a limit.

---

Intuition for Calculus — What is a Limit?

Here is a thoughtful question: What if we arbitrarily convert an infinite series into a finite series by specifying a final term? Such a sum up to any specified term in an infinite series is called a partial sum.

Let us now say that we are going to keep cycling through the final term such that we move forward in the series. What happens to the corresponding partial sums?

As we continue cycling through the final terms of an infinite series, the corresponding partial sums get closer and closer to a number, say, ‘L’.

By continuing the series as much as we need, we may get closer and closer as we wish toward ‘L’. This ‘L’ is known as the limit of the infinite series. Consequently, the terms ‘converge’ on the limit ‘L’. If the terms are not converging, then they are ‘diverging’.

A more intuitive definition of the limit of an infinite series is that it is the series’ sum at infinity. Generally speaking, an infinite series has three convergence possibilities on its limit:

1. The partial sums get ever so close to the limit without actually ever reaching it or exceeding it.

2. The partial sums converge exactly on the limit.

3. The partial sums shoot beyond the limit before they converge.

To understand how these work, let us take a look at a few examples.

Zeno’s Paradox on What is a Limit

Zeno, from the fifth century B.C., Greece, invented numerous famous paradoxes that dealt with our seemingly poor intuition around motion. One such paradox serves us well with the notion of a limit.

Zeno assigned a runner running from point A to point B. Zeno’s runner first runs half the total distance between A and B. Then, he runs half of the remaining distance, after which, he yet again runs half of that remaining distance, and so on.

The distances covered by the runner get smaller and smaller at a constant proportion as given by this unique series:

Distance covered by Zeno’s runner = 1/2 + 1/4 + 1/8 + 1/16 +…

It is interesting to note that the distance the runner has covered from A converges on 1 and the distance from B converges on zero. Having said this, does the runner finally reach point B?

Well, the answer depends on how the runner runs, and what we mean by ‘finally’.

---

Tackling Zeno’s paradox

Let us say that Zeno’s runner pauses for one second after covering half the distance between A and B. Then, after covering half of the remaining distance, he pauses for another second. Similarly, he keeps pausing for a second each time he halves the distance.

Here, the runner’s pauses make the problem a function of a discrete variable. Consequently, by continuing so, Zeno’s runner will get closer and closer to point B, but will never reach it.

However, many mathematicians would argue that he would get close enough to point B for almost all humanly practical purposes.

Let us now switch things up and say that Zeno’s runner is running at a constant rate without pauses. Say that he reaches half distance between point A and point B in one second. Then, he would reach half of the remaining distance in half a second, and so on.

In this scenario, the problem becomes a function of a continuous variable. In other words, we have transformed the problem from a discrete domain to a continuous domain. Furthermore, in this case, the time taken for Zeno’s runner to reach point B converges exactly on the limit.

“What exactly is this limit?”, you ask? Well, let us compute it. And I even have a neat little trick to show you whilst doing it:

So, as you have just seen, in the discrete domain, Zeno’s problem leads us to a series that gets ever so close to the limit, but never reaches (or exceeds) it. And in the continuous domain, the problem leads us to a series that converges exactly on the limit.

There is one more convergence possibility that we have not covered yet.

A Series that Shoots Beyond the Limit Before it Converges

Let us now alter the above halving series by changing every alternate positive sign to a negative sign as follows:

1/2–1/4 + 1/8–1/16 +…

If you compute partial sums of this series as you cycle through the series, you would see that the sums “alternate”. That is, the partial sums alternate around the limit.

You can reduce the absolute difference between the partial sums and the limit by choosing terms farther and farther into the series. However, the alternating behaviour does not go away.

Consequently, this series is an example of one that shoots above (and below) the limit before it converges on the limit.

You might have noticed that I haven’t mentioned what the limit for this series is or how it exactly converges. If you are interested in a little exercise, you might try and work out the limit of this series and figure out its nature on your own.

Closing Comments

We just covered the notion of a limit using partial sums of infinite series. While the notion of a limit is critical to proceed with conventional calculus, it is by no means mandatory.

There exists another variant of calculus called Nonstandard calculus (Nonstandard analysis is the mother term), where one uses infinitesimals and not limits.

I find infinitesimals a tad bit more intuitive than limits. However, they are both trying to achieve the same goal.

I will be covering the ground fundamentals of the infinitesimal approach to calculus in my next essay as part of the calculus series.

If you are interested in it, keep an eye out. Or better yet, consider subscribing!

---

Reference and credit: Martin Gardner.

Further reading that might interest you:

• Is ‘0.99999…’ Really Equal To ‘1’?
• Can You Really Solve The Staircase Paradox?
• What Really Happens When You Invent Infinite Infinities?

If you would like to support me as an author, consider contributing on Patreon.