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Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

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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.

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.

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**.

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?

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.

I can graph that another way. If I graph the location as a function of time, my graph approaches the location at *t*=4 from both the left-hand side where time is increasing and the right-hand side where time is decreasing. I could say that the limit of the position as we approach the time of 4 is here.

Let's look at some mathematical examples. Let's look at the function the absolute value of *x*, or |*x*|. The graph of |*x*| looks like a 'V', and you know that as you approach the value *x*=0, *f(x)* will equal 0. Say *f*(0.1)=0.1, *f*(0.01)=0.01, *f*(0.001)=0.001, *f*(0.00000001) is ... well, you get the idea. If we approach it from the other side, we get the same behavior. *f*(-0.1)=0.1, *f*(-0.01)=0.01 and so on and so forth. In both cases, the value of this function is approaching zero as we go to *x*=0.

What about a function that is not continuous? So for a function that is not continuous, such as *f(x)*=*x* for *x* less than or equal to zero, and *f(x)*=1 for *x* greater than zero, as we approach *x*=0 from the left-hand side, our function approaches the value of zero. As we approach *x*=0 from the right-hand side, our function approaches the value of 1. This might be something to consider a little later on. What about the function, *f(x)* = 3(*x*)^2? Well, there's a limit at, say, *x*=2.

So really, **limits** tell you what happens to functions as you approach some value. For road trips, that may be pedal to the metal as an asymptote. For pendulums though, this is what happens as you approach some given point in time. For graphs, we look at this as what happens to your function as you approach some value of *x*.

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Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

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