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What Is a Mathematical Limit?

Do you know what a limit is in math? Do you know how to define a circle using this idea? And do you know why you might want to? Keep on reading to find out!

By
Jason Marshall, PhD,
October 10, 2015
Episode #260

Page 1 of 2

DecagonIn everyday language, the word "limit" is used to describe the boundaries beyond which some quantity or some idea or some thing can't exceed. For example, the speed limit tells you the maximum rate you're legally allowed to drive. And your credit card limit tells you the maximum balance you can carry. Both of these quantities represent upper boundaries. Of course, limits can also apply to lower boundaries. For example, the GPA cutoff put in place by a college's admissions committee or the minimum credit score required for a loan.

In math, the idea of a limit is kind of the same … but it's also kind of different. It's the same in that a limit is used to talk about what happens as you get closer and closer to some condition or boundary. But it's different in that it isn’t necessarily about the minimum or maximum values associated with these things. Instead, in math, the idea of a limit and the type of boundaries it deals with can be much more abstract.

So, how do limits in math work? And why do they matter? We're about to find out.

How to Define a Circle

In order to understand the mathematical definition of a limit, let's talk about the mathematical definition of a circle. In particular, let's talk about the various ways in which we can precisely define what we mean by a circle. First up, there’s the geometric way in which we say that a circle is a two-dimensional curve (meaning it lives on a flat sheet of paper) in which all the points along the curve are located the same distance away from some central point. And then there's the algebraic way of defining a circle in which we say that a circle is the shape made from all the points in the x-y plane that are solutions to the equation x2 + y2 = r2 (where r is the radius of the circle). Of course, these two definitions are related—in fact, they’re two different ways of saying the same thing.

As you draw regular polygons with more and more sides, do you notice anything happening to the shape?

While both of these are fine and dandy ways of defining a circle, I know another one that I think is more interesting. Here’s how it works: Start by drawing (or start by imagining drawing) an equilateral triangle (also known as a three-sided regular polygon—the word “regular” here tells you that the sides of this polygon are all the same length). Next to this triangle, draw a square (also known as a four-sided regular polygon). Next up in the series, draw a five-sided regular polygon (aka, a pentagon), then a six-sided regular polygon (a hexagon), a seven-sided regular polygon (a septagon or heptagon depending on who you ask), and on and on. As you draw regular polygons with more and more sides, do you notice anything happening to the shape? Do you notice that it's starting to resemble something … perhaps a circle?

What Is a Limit in Math?

If you think about it, you’ll see that we could continue this procedure of systematically drawing regular polygons with one more side than the previous indefinitely. I don’t have enough patience or, more importantly, time to do that (knock yourself out though), which means we’re going to stop our drawing extravaganza with our seven-sided shape. But the fact that we could continue forever is important. In fact, it’s the key to understanding the mathematical idea of a limit.

Because here’s the kicker that you may have already noticed: As we increase the number of sides in our regular polygon, the shape we draw begins to look more and more like a circle. We stopped at seven-sides, but you can imagine drawing a 10-sided regular polygon … or a 50, 100, 1 thousand, or 1 million-sided regular polygon. The more sides, the closer and closer the shape gets to a perfect circle. In the beautiful language of math, we say that the shape drawn will become a circle in the limit that the number of sides of our regular polygon goes to infinity.

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