2 LIMITS

SECTION 2.2THE LIMIT OF A FUNCTION• Stress that if lim f (x) = ∞ and lim g (x) = ∞, then we still don’t know anything aboutx→a x→a[ ] f (x) − g (x) .limx→aWORKSHOP/DISCUSSION• Explore the greatest integer function f (x) = [[x]] on the interval [−1, 2] in terms of left-hand and righthandlimits.(• Explore lim 2 − 21/x ) , looking carefully at 2 1/x from both sides for small x.x→0numerical, and algebraic (“what happens to 1/x when x is small?”) contexts.Discuss in graphical,x − 3• Discuss a limit such as lim . When can you “just plug in the numbers”?x→3 4x + 2GROUP WORK 1: Infinite LimitsAfter the students are finished, Problem 2 can be used to initiate a discussion of left and right hand limits,and of the precise definition of a vertical asymptote, as presented in the text. In addition, Section 4.4 canbe foreshadowed by asking the students to explore the behavior of 3x2 + 4x + 5√ for large positive and large16x 4 − 81negative values of x, both on the graph and numerically. If there is time, the students can be asked to analyzethe asymptotes of f (x) = sec x and the other trigonometric functions.Answers:1. Answers will vary. The main thing to check is that there are vertical asymptotes at ± π 2 and at ± 3π 2 .2.y20There are vertical asymptotes at x =±1.5.10_4 _2 0 2 4x43