Limits and Continuity ( 53 Example 1 Given f(x) = * - 2x - 3 x - 3- find the limits: (a) lim f(x), (b) lim f(x), and (c) lim f(x). Substituting x = 3 into f(x) leads to a 0 in both the numerator and denominator. Factor f(x) as (x - 3)(x + 1) -, which is equivalent to (x + 1) where x / 3. Thus, (x - 3) * -+ 3+ (a) lim f(x) = lim (x + 1) = 4, (b) lim f(x) = lim (x + 1) = 4, and (c) since the x-+3+ one-sided limits exist and are equal, lim f(x) = lim f(x) = 4, therefore the two-sided x-+3- x- 3 limit lim f(x) exists and lim f(x) = 4. (Note that f(x) is undefined at x = 3, but the * +3 function gets arbitrarily close to 4 as x approaches 3. Therefore the limit exists.) (See Figure 5.1-3.) [-8, 8] by [-6, 6] Figure 5.1-3 Example 2