Understanding the Properties of Limits

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  • 0:06 Properties of Limits
  • 1:14 Addition and…
  • 2:32 Product Property
  • 3:26 Division Property
  • 3:56 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

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

Graphically we can see limits, but how do we actually calculate them? Three words: Divide and Conquer. In this lesson, explore some of the properties that we can use to find limits.

Properties of Limits

Let's say that we need to find the limit of f(x) as x goes to some number, like 3. Recall that a limit is what f(x) is going to approach as x goes to 3. This limit can be one or two-sided. If you go from the left side and it's equal to one thing, and if you go from the right side and it's equal to something else, then it's a one-sided limit. If these two values are the same, then it's a two-sided limit.

The addition and subtraction properties of limits
Limit Addition Subtraction Property

Let's try to find the limit of f(x) = 3x^2 - 1. Well, I can just graph that to try to find the limit as x goes to something like 3. What about something like f(x) = 3(x^2 - 1)(x + 1)^2(3x - 4x^(-1))sin(x)tan(x) … Other than graphing it, or calculating numbers explicitly, how can we find the limit?

Addition and Subtraction Properties

Before we actually get into finding the limit, let's take a look at some of the properties that might help us find limits. The first properties of limits are fairly straightforward. These are the addition property and subtraction property. They say that the limit of some sum, like f(x) + g(x), as x goes to some number, like 3, is equal to the limit, as x goes to that number, 3, of f(x) plus the limit, as x goes to 3, of g(x). You've got the first function plus the second function. So this means that we can find the limits separately and just add them together.

Let's consider our function h(x) = x + 3. What is the limit, as x goes to 3, of h(x)? I can use the property of addition to say that the limit of x + 3 as x goes to 3 is equal to the limit of x as x goes to 3, plus the limit of 3 as x goes to 3. In this case, I'd find that the limit is 6.

The product property
Limit Product Property

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