What is the Limit of a Function

The limit of a function may be a confusing concept. What do we mean when we say that a function has a limit at a given point? And how can we estimate or compute limits? Limit problems show up on both the AP Calculus AB and BC exams, so its important to understand the concepts and techniques in order to maximum your score. In this article Ill define the limit of a function and illustrate a few techniques for evaluating them. Definition: the Limit of a Function Suppose y = f(x) is a function. Informally, a limit of f is a y-value L that f(x) approaches as x approaches some specified number a. We use the following notation for limits: Graphical Interpretation of Limits Lets take a look at the graph below. What is going on near x = 3? The open circle on the graph at (3, 2) means that f(3) does not exist. However the limit does exist! Notice that any x-value in the green region has a corresponding y-value in the blue region. As long as x is chosen close enough to 3 (but not 3 itself), the output of f(x) can be made to be as close as we want to 2. Therefore the limit of f(x) as x 3 is equal to 2. Estimating a Limit I tend to think of limits as measuring the trend of the function values. One way to estimate a limiting value is to build a table of values. Choose x-values that are closer and closer to x = a, and look for a trend in the corresponding y-values. For example, lets estimate , using a table of values. First of all, note that if you plug in x = 3, you get: (32 3(3))/(32 9) = (9 9)/(9 9) = 0/0. We all know you cant divide by 0. So does that mean the limit doesnt exist? No, the limit might still exist! In fact, a limit problem is not really asking for the actual value of f at x = a. Instead, its about what happens near x = a. So since we cant plug in 3 directly, we should examine the output of the function for x-values near 3. For example, try plugging in 2. But there are infinitely many numbers that are even closer, including 2.9, 2.99, 2.999, so lets check those out too. xf(x) = (x2 - 3x)/(x2 - 9) 20.29 3.10.508197 3.010.500832 3.0010.500083 3Can't plug in Even though you cant plug in 3, the trend in the function values seems to be point towards a limiting value of 0.5. Next Steps Finding a limit from the graph or from numerical evidence may not give the precise limit value. In fact, a table of values may even mislead you to think that a function has a limit when it does not. So its essential to have a toolbox of methods for working out limits analytically. In this short introduction we cannot explore those techniques, but now that you know what a limit is, take a look at the following resources: AP Calculus Exam Review: Limits and Continuity Limits on the AP Calculus Exam.