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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Join me on a road trip as we define the mathematical notation of limits. As time goes by and I traverse hills and highways, the limit of my speed changes. Learn how to write these limits in this lesson.

Defining Limits

Roadtrip graph showing speed as a function of time

Limits describe what happens as you approach some number. But how do we write this mathematically? Let's go back to a road trip of mine. I'm going to plot my speed as a function of time. I'm going to start out and drive along the city streets. Then I'm going to get on the highway and cruise along at 60 mph or so. I'll go over a hill where my car slows down. When I go down the hill, I start going too fast. Then I'm going to get pulled over for speeding. I'm going to say that when I get pulled over for speeding, that's about at a time roughly 40 minutes in. So I'm going to say that my time, which I'll call t, is 40. How do we write the limit as I approach one of these times? So what is the limit? The limit is the value that a function approaches as your variable approaches some number. In this case, the variable of interest, the variable I care about, is time. Our function is speed.

What is the limit that my speed takes as I approach 10 minutes? Well, if I'm on a highway, I'm probably going about 60 mph. Mathematically, I'd write the limit as time goes to 10 of my function's speed, which is a function of time, is 60. Similarly, I could say that the limit as time approaches 20 of my speed is about 45 mph. We know that when I go uphill, I can only get up to about 45 mph. Lastly, I can say that the limit as time approaches 30 of my speed is maybe 70 mph. This is what I was going when the cop found me.

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Limits in math are just like in the real world. They're what happens as you approach something. Say you want to find the limit of f(x) as x approaches 2. One way you can do it is to look at what happens to values of x as you approach 2. Let's say that at 2, f(x) is undefined. But what happens as we approach 2 from, say, the left-hand side? As we approach 2 from the left-hand side, f(x) seems to be moving toward 4. If we approach it from the right-hand side, for this particular function, it also approaches 4. So we might write that the limit of f(x) as x goes to 2 is 4.

You can do this for a lot of functions. For example, f(x) = 10(7 + x)^(1/3) for x does not equal 1 and f(x) = 1 when x = 1. What is the limit as this function approaches 1? Let's write this out in a horizontal table. At zero, f(x)=19.13. As we increase x, f(x) is also increasing. It looks like f(x) is approaching 20.

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