Finding Limits in the wild: Why Engineers Should Care About Limits

After reading this blog a couple of times, you may ask:

Well, Jasper. You say you like math and science, then what do you do for a living?

Well, I am a PhD-student and work in the field of control theory and dynamical systems. What I exactly do is surely something I want to talk about in the future, but it might lead us too far for now.

Since I am a PhD-student, I also have some teaching duties. Hence, I am giving the exercise sessions of the calculus course in the engineering science program. These exercise sessions are really nice because it provides some variation during the week. I can dive in the math I also had to study and it's nice to go back to the roots. In addition, these exercise sessions provide good suggestions for blog posts.

When teaching calculus, one of the most common, and honest, student questions goes something like this:

Why do we need to learn about limits and all these messy rules like L'Hôpital's and Taylor series? How will I ever decide if a bridge collapses by calculating a limit?

And that's a fair question. After all, in the real world, you usually won't sit down and compute $$ \lim _{x \rightarrow 0} \frac{\sin x-x}{x^{3}} $$ But, this mindset misses one thing. You couldn't even write the equations that predict whether a bridge breaks without the idea of a limit. Many equations in structure mechanics are based on the idea of a limit. When you analyze a very small part of a structure, you're effectively taking the limit as that part's size goes to zero. This is done A LOT in structure mechanics. Many things in science, start by saying,

What if we consider a small part of something?

or

What if we give something a very little push?

Nature is sneaky. Everything around you is made of a discrete number of particles: atoms, electrons, digital pixels. However, everything looks smooth. A river looks to be a continuous flow. Why? Because when billions of tiny things act together, everything blurs into a continuous flow. That blur is what limits formalize.

So, okay, limits are apparently important, but still you are not convinced. Let me give you a short tour to show explore why limits matter and are actually one of the deepest ideas in all of mathematics and engineering. For this blog, we are not going into the mechanics of how to actually compute them, but we will discuss why the idea of limits is useful.

Today, we go look for limits in the wild and find out their purpose.

Limits are the foundation of calculus

Some time ago, we briefly discussed limits in the post about Heron's method. However, why do we care? Well, When you sit down and list every important concept in calculus, then it turns out that EVERY SINGLE ONE OF THEM is based on a limit.

Considering many small parts of a structure, you can simulate how a complete structure behaves under stress and strain.

Then let's consider the first large pillar of calculus: derivatives. The derivative of a function $f$ is the instantaneous rate of change in a given point of the curve of that function $f$.

Imagine you're hiking up a mountain with the height of the mountain described on point $x$ as $f(x)$. Over some horizontal distance, you can compute your average rate of climb. In other words, you can determine the height increase $\Delta y$ on average over a horizontal distance $\Delta x$. For a height of $f(x)$ the average rate of change between the points $x_{1}$ and $x_{2}$ is: $$ r=\frac{\Delta y}{\Delta x}=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{\Delta x} $$ For which $\Delta y$ is the vertical increase of distance when walking a horizontal distance $\Delta x$. However, how can you correctly define the rate of increase when you stand at a specific point like $x_{0}$ ?

Now, we do not need to compute the average over a walked horizontal distance $\Delta x$, but to compute the rate of change at one specific point. How can we do this actually? The trick is to make $\Delta x$ very small. Consequently, we need a limit and that limit is $$ f^{\prime}\left(x_{0}\right)=\lim_{\Delta x \rightarrow 0} \frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{x_{0}+\Delta x-x_{0}}=\lim_{\Delta x \rightarrow 0} \frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{\Delta x} . $$ That’s the moment calculus was born. Someone, somewhere, asked:

What if I shrink my step $\Delta x$ until it disappears?

WAIT A MINUTE! We make $\Delta x$ very small, don't we divide by zero then? And we do know for a fact that dividing by zero is not defined. Well, assuming that you are not walking on a mountain which has a vertical slope you are okay. If this is not the case then the change of $f$ for a small step $\Delta x$ will also be very small. This means that you get an indefiniteness because you have a zero in both the numerator and denominator and usually, these zeros will cancel out if your mountain does not have any vertical slopes or has sharp edges.

For properly saying what we mean with a derivative we need to use limits. We cannot say:

Trust me bro, we make $\Delta x$ arbitrarily small.

We need to actually properly say what we are doing and we can do so be using a limit.

Do you know what also uses limits? It is the other thing you discuss in calculus. It is usually everyone's favourite. CORRECT: integrals! Suppose that you want to estimate the area under a curve $f$ then you can just go for the engineering approach. Slice the area in different intervals and approximate the actual area by a bunch of rectangles.

In other words, the more rectangles we take, the closer we get to the actual area of the area. This also looks like: $$ A=\int_{a}^{b} f(x) d x=\lim_{N \rightarrow \infty} \sum_{i=1}^{N-1} f\left(x_{i}\right) \Delta \mathrm{x} $$ $$ x_{i}=a+(b-a) \frac{i}{N}, \quad i=1, \cdots, N-1 $$ $$ \Delta x=\frac{b-a}{N} $$ Again, we simply need a limit for properly describing what we are doing.

A small disclaimer: when actually computing integrals in practice, we often use more sophisticated methods. But the Riemann sum is the conceptual foundation. Nonetheless, the Riemann sum is the most important integral interpretation and is crucial in calculus.

So in essence, literally everything we do in calculus is built on the notion of a limit. Without them, we couldn't define what it means for something to change smoothly, become very small, very large, ...

Before the concept of a limit was formalized, approaching a value was just intuition. If you think that this concept of a limit is weird, don't worry. The ancient Greeks also struggled with the idea of something becoming arbitrarily small or large. Democritus reasoned that if you repeatedly divide a substance, you must eventually reach a point where it can no longer be divided. Zeno's paradox, which is something for a future post, is another illustration of the Greeks struggling with limits. However, to discover new concepts you sometimes need to drive against the current ideas. As limits are essential to everything in modern math it is safe to say that if the ancient Greeks already understood limits then current math would be much more advanced.

Bridge between the discrete and the continuous

Matter is made of atoms, signals are transmitted by electrons, computers handle numbers one step at a time. So in essence, everything is guided by many discrete things. But models in science and engineering are continuous. A beam bends smoothly, a fluid flows continuously, a voltage varies over time. When many discrete components act together, their combined behaviour appears continuous.

The best example I have seen of this, is that you have many billiard balls with a constant speed and that they can collide with each other. If you make the amount of billiard balls very large then you get to see very continuous behaviour like a diffusion of particles in a space. Other examples is the world of finance where a large amount of discrete actors make the market look like a continuous moving thing.

By adding up many discrete effects, you end up with something that behaves continuously. It's actually similar to plotting a curve $f(x)$ on a computer. What you see are discrete sampled points that appear smooth.

But then, how do we justify replacing the discrete with the continuous? By taking a limit we can go from a discrete world to a continuous. Just as we increase the amount of billiard balls we get a continuous flow. In the same way, a sum becomes an integral. $$ \int_{a}^{b} f(x) d x=\lim_{N \rightarrow \infty} \sum_{i=1}^{N-1} f\left(x_{i}\right) \Delta \mathrm{x} $$ A finite difference between two function values becomes a derivative $$ \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=f^{\prime}(x) $$ Going from the discrete to the continuous world is one of the great conceptual steps in all of science and it's powered entirely by the idea of a limit. In engineering, we simply need limits as they turn that intuition into a mathematical statement. Many commonly said things in engineering are:

As time goes to infinity, the system stabilizes.

The functions $sin(x)$ and $tan(x)$ can be accurately approximated by $x$ around the point $x = 0$.

Each of these statements is really about a limit, the tool that lets us reason about behaviour that is almost at a value, or infinitesimally close. The idea is at the heart of every approximation we make.

Limits are the language of approximation

Something what engineers love, is building an approximation of a complex model. Usually this is done by linearizing a system or neglecting small terms. Again, a limit is at the base of this idea. The most famous example is the so-called small-angle approximation which states that $\sin (\theta) \approx \theta$ as $\theta \rightarrow 0$.

If you know your limits then this is just the same as saying that $$ \lim _{\theta \rightarrow 0} \frac{\sin (\theta)}{\theta}=1 $$ This approximation only holds around $\theta=0$ and can only work when your angle $\theta$ is small. I must say THAT FOR THE LOVE OF GOD, you should not use this approximation for $\theta=50$ radians because this value is far away from zero and the approximation will be BAD. Unless you enjoy exploding bridges and deeply disappointed physics teachers. However, if you want to fool around with non-engineers I always invite you to casually state that $e=\pi = \sqrt{g}$.

This is all to say that techniques to compute limits like a Taylor series, L'Hôpital's rule, and other tools are not arbitrary tricks. They provide the foundation on how to approximate behaviour near a point, giving us the power to turn difficult problems into manageable ones.

How to find limits in the wild?

Ok, this is very nice and all, but if limits are something theoretical. Why should an engineer care about them? Imagine you're studying something like an electric car motor.

With limits, you can explore how the motor behaves in extreme situations. For example, what happens if a certain setting becomes very large or very small. You can also use limits to see how the motor reacts over time: how quickly it reaches its top speed, or how it settles down to a desired speed. And if you slightly change one of the motor's parameters, limits help you predict how that small change affects the overall behaviour.

When studying an electric motor you can still apply the principles of limits to make your life as an engineer easier.

However, it doesn't stop here. To make your life as an engineer easier, you can approximate the complicated motor model around a preferred operating point, like a preferred motor speed, so that you have a simplified model which mimics the original model. Then limits help you to determine which terms matter, which can be ignored. Essentially, you can apply this idea to anything. A self-driving car, a chemical reactor, an autonomous robot, an rocket ship ... This mindset is everywhere in engineering.

Limits are not about calculating. They're about understanding behaviour, how things change, approach, and stabilize. If calculus is the language of change, then limits are its grammar. They hide in the shadows, but you actually use them daily every time you build, design, or explain how something changes.