LIMITS AND CONTINUITY

 The concept of the limit in calculus is very important. It describes what happens to a function as a particular value is approached. The derivative, one of the major themes of calculus, is defined in limit terms. This short chapter will help you to think in terms of limits. The first thing to understand about limits is that a limit of a function is not the value of the function. The change in thinking (from value to limit) is important because most functions are understood as a series of mathematical operations that can be evaluated at certain points simply by substitution.

The (polynomial) function y = x² + 2x + 3 can be evaluated for any real number: replace x with the number and perform the indicated operations. Asking the limit of this function as x approaches 2, for example, is an uninteresting question. The function can be evaluated at 2 or any point arbitrarily close to 2 by substituting and performing the operations.

Other functions, such as polynomial fractions, cannot be evaluated at certain points and these functions are best understood by thinking in terms of limits. The function y = (x² – 4)/(x + 2) can be evaluated for any real number except –2. Replacing x by –2 produces the meaningless statement 0/0. Remember that any number times 0 is 0, but any number divided by 0 is "meaningless" (including 0/0). Looking at the limit of the function, as x approaches –2, tells us about the function in the vicinity of –2. The limit of the function is a convenient phrase for the question, "What happens to the function as a certain value is approached?" Writing this in mathematical notation we get the following:

 The notation in front of the functions is read "the limit, as x approaches minus two." In the case of rational functions, factoring and reducing the fraction helps in finding the limit.

 Finding the limit of this function as x —> –2 helps in understanding the function. Since the original function gives the meaningless 0/0at the point where x = –2, the function cannot exist, "does not have meaning," at x = –2. Graphing the function illustrates this point. The (simplified) function y = x – 2 is a straight line of slope 1 and intercept –2. The function y = (x² – 4)/(x + 2) is also a straight line of slope 1 and intercept –2, but it does not exist at the point where x = –2. This non-existence at x = –2 is illustrated on the graph in Fig. 2-1 with the open circle.