You will find a graphing calculator useful for Exercises 67–76. Let ƒ(x) = (x2 – 9)/(x...

You will find a graphing calculator useful for Exercises 67–76.

Let ƒ(x) = (x² – 9)/(x + 3).

a. Make a table of the values of ƒ at the points x = -3.1, -3.01, -3.001, and so on as far as your calculator can go. Then estimate What estimate do you arrive at if you evaluate ƒ at x = -2.9, -2.99, -2.999, … instead?

b. Support your conclusions in part (a) by graphing ƒ near c = -3 and using Zoom and Trace to estimate y-values on the graph as

c. Find algebraically, as in Example 7.

EXAMPLE 7 Evaluate

Solution We cannot substitute x = 1 because it makes the denominator zero. We test the numerator to see if it, too, is zero at x = 1. It is, so it has a factor of (x – 1) in common with the denominator. Canceling the (x – 1)’s gives a simpler fraction with the same values as the original for x ≠ 1:

Using the simpler fraction, we find the limit of these values as by substitution:

See Figure 2.11.