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
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
The second property of limits is the product property. This says that the limit, as x goes to some number, like 3, of a product or a multiplication of two functions, like f(x) * g(x), is equal to the limit as x goes to 3 of f(x) all times the limit as x goes to 3 of g(x).
So if you have a function like x^2, the limit as x goes to 3 of x^2, is just like saying the limit as x goes to 3 of x, times the limit as x goes to 3 of x. This is because we've divided our x^2 into x * x. And in this case, if you graph it out, you'll see that it's 9, because the limit as x goes to 3 of x is just 3. And 3 * 3 = 9.
The final property is the division property. Here, if you're trying to take the limit of one function divided by another function, it's equal to the limit of the top function divided by the limit of the bottom function.
The division property
So if you have the limit as x goes to 3 of f(x/3), that's like saying the limit as x goes to 3 of x, divided by the limit as x goes to 3 of 3, which is just 1.
The big thing to remember with limits and properties of limits is to 'divide and conquer'. Take whatever function you are trying to find the limit of and divide it up into its individual parts. Then use the properties of limits to find the overall limit. If you have a really nasty function, look at its individual parts and find the limits of its individual parts. Then put them all together to find the limit of f(x).