Using a Graph to Define Limits

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  • 0:06 The Limits of My Car
  • 1:02 Examples of Limits
  • 1:29 Limits of a Pendulum
  • 3:07 Examples of Limits in Math
  • 4:58 Lesson Summary
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Lesson Transcript
Instructor: Zach Pino
My mom always said I tested the limits of her patience. Use graphs to learn about limits in math. You won't get grounded as we approach limits in this lesson.

The Limits of My Car

I've been on a few road trips moving across the country, and during every single road trip, I hit mountains. Now I don't know about you, but when my car hits the hills, it's always an adventure. No matter how hard I press the gas pedal, my car never goes faster than about 45 miles per hour. The hill just seems to be limiting my speed.

Graph of the pendulum location as a function of time
Pendumlum limits

So let's graph this. Say I graph speed as a function of the pedal depth. Say my speed is 1 mile per hour because cars never quite stop. As I push harder on the pedal, my speed increases, increases, increases, but if I'm at a hill, I get to about 45 miles per hour. No matter how hard I push, I just can't break through 45 miles per hour. Now we know that this is a horizontal asymptote, but we can also call it a limit.

Examples of Limits

There are limits all over the place. From, obviously, my car on a hill, to the terminal velocity when you jump out of a plane and are skydiving. You're going to hit some velocity and not go any faster than that; you are limited. My mother's patience is limited, and so on. But mathematically, we look at limits as functions as we approach, say, a number. So what does this mean?

Graph of the function |x|
Limits in math example 1

Limits of a Pendulum

Let's consider a pendulum. So a pendulum, let's say your watch - you're trying to hypnotize somebody - is swinging back and forth along a string. Now I can take a look at the location along x as a function of time, and maybe when I release it from zero, the time here is zero. The pendulum goes through and makes one swing, and on the other side the time is about 6. So if I graph the location of the pendulum and mark out the time that it's at a particular location, I might get something that looks like this. As we move forward in time, we know that the location of the pendulum will follow this line, and at t=4, we expect it to approach and reach this point. If we start at 5 seconds in, say here, we can reverse time, and if we reverse time, we see that the pendulum will approach the same point where it was at t=4. The limit of the pendulum location is right here at t=4.

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