Roots and Powers of Algebraic Expressions

Subtraction Property and Limits: Definition & Examples

What are limits of functions? What happens to limits if we subtract two functions that have limits at the same value? In this lesson, we will look at the subtraction property for limits.

Definition of a Limit

Most of the time, a function f(x) has a value for every possible value of x, but not always. For example, f(x) = (x - 4)(x - 6)/2(x - 6) is undefined at the value x = 6 because dividing by 2(6- 6) = 0 is just not feasible. Even then, however, we can look at what the function is like when we get closer and closer to the limit: in this case, the limit when x = 6.

As we can see below, the closer x gets to 6, the closer y gets to 1, so we say that the limit of f(x) as x approaches 6 is 1. We can define a limit if we can get as close as we want to a particular value of x (call it x sub 0) and if it consistently gets closer and closer to one particular value of y when we do. If we can't get close to the x value with a definition of a function, or if the values of y jump around when we get closer and closer to x sub 0, or if the values of y go up or down without ever settling near a number, we cannot define a limit.

Subtraction Property

The subtraction property for limits says that, if two functions, say f(x) and g(x), have a limit at the same point (let's call it a), the limit as x approaches a of (f - g)(x) is the difference, as x approaches a, of the two limits at a. Note that both limits need to be approaching the same number.

Now, suppose we have two functions, and both have limits at a certain value. For example, suppose f(x) = (x^2 - 1)/(x + 1) and g(x) = x + 3. Each of them has a limit at x = -1: the limit of f(x) as x approaches -1 is -2, and the limit of g(x) as x approaches -1 is 2. Let's look at the graphs:

So far, so good. Notice that g(x) has a defined value at x = -1 and f(x) does not. The limits still work out just fine.

Now, suppose we look at (f - g)(x). We can subtract most functions by subtracting the expressions that define them, so in our example,

(f - g)(x) = (x^2 - 1)/(x + 1) - (x + 3) = (x^2 - 1)/(x + 1) - x - 3.

Let's graph the new function and see what happens to the limit as x approaches -1:

Look! The function turned out to be almost a constant function, with every value except (f - g)(-1) coming out a -4. Of course, (f - g)(x) is not defined for x= -1.

So the limit as x approaches -1 for f(x) is -2, the limit as x approaches -1 for g(x) is 2, and the limit as xapproaches -1 of (f - g)(x) is -4, which is -2 - 2.

We can subtract the two limits to get the limit of the difference function and save some work. This is the Subtraction Property for limits:

The limit as x approaches some value a of (f - g)(x) is equal to the limit as x approaches a of f(x) - the limit as x approaches a of g(x), providing that both limits are defined.

Summary

When we have a limit defined at a certain value a for two functions f and g, the limit at that value for the difference function (f - g)(x) is the difference of the limits for f(x) and g(x) at that value. Instead of defining the difference function and finding the new limit, we can just subtract the known limits. Note that, if the value of either function is undefined at a, the value of the difference function is also undefined at a.