(9/30/08)Math 10A. Lecture Examples.Section 1.8. Limits†Example 1 Figure 1 shows the graph o f the function,F(x) =(4 − x2for x < 14 for x = 12x for x > 1.(a) Calculate F(x) at x = 0.9, 0.99, 0.999, and 0.999, 1.1, 1.01, 1.001 and1.0001. (b) What is limx→1−F(x)? (c) What is limx→1+F(x)?x−1 1 2y123y = F (x)4FIGURE 1Answer: (a) The values are in the table below. (b) limx→1−F (x) = 3 • Figure A1a (c) limx→1+F (x) = 2 •Figure A1bxF (x) = 4 − x20.9 3.190.99 3.01990.999 3.0019990.9999 3.00019999x F (x) = 2x1.1 2.21.01 2.021.001 2.0021.0001 2.0002†Lecture notes to accompany Section 1.8 of Calculus by Hughes-Hallett et al.1Math 10A. Lecture Examples. (9/30/08) Section 1.8, p . 2x−1 1 2y3y = F (x)xy = 4 − x2x−1 1 2y2y = F (x)xy = 2xlimx→1−F (x) = 3 limx→1+F (x) = 2Figure A1a Figure A1bExample 2 Figure 2 shows the graph o f y = sin(1/x) for x > 0. Explain whylimx→0+sin(1/x) does not exist.x0.5y1−1y = sin(1/x)FIGURE 2Answer: Since sin x oscillates infinitely often between 1 and −1 as x increases through all positive values, sin(1/x)oscillates infinitely often between 1 and −1 and do es not approach any one number as x approaches 0 from the right.Example 3 What is limx→5+A(x)B(x) +C(x)D(x)if limx→5+A(x) = 2, limx→5+B(x) = 5,limx→5+C(x) = 6, and limx→5+D(x) = 3?Answer: limx→5+A(x)B(x) +C(x)D(x)= 12Section 1.8, p . 3 Math 10A. Lecture Examples. (9/30/08)Example 4 Draw the graph of J(x) =x2for x < 28 − 2x for x > 2and find the limit of J(x)as x → 2.Answer: Figure A4 • limx→2J(x) = 4x−2 2 4y−224y = J (x)Figure A4Example 5 What is limx→1x + 5x + 2?Answer: limx→1x + 5x + 2= 2Example 6 Find limx→1T(x) whereT(x) =x + 1 for x ≤ 11/x for x > 1.Answer: limx→1T (x) is n ot defined (does not exist).x1 2 3−1y12y = T (x)Figure A6Math 10A. Lecture Examples. (9/30/08) Section 1.8, p . 4Example 7 Figure 3 shows the graph o f a function K, defined byK(x) =x + 4 for −2 ≤ x < 1x + 1 for 1 ≤ x ≤ 4.(a) At what values of x for −2 ≤ x ≤ 4 is K continuous? (b) Is K continuousfrom the le ft or from the right at the values of x for −2 ≤ x ≤ 4 where it is notcontinuous? (c) What are the largest intervals on which K is continuous?†x−1−2 1 2 3 4y1235y = K(x)FIGURE 3Answer: (a) K is continuous at all x with −2 < x < 1 and 1 < x < 4. (b) K is not continuous at −2(because it is not defined for x < −2) but is continuous from the right at −2. • K is not continuous at 1 (becausethe one-sided limits are different) but is continuous from the right at 1. • K is not continuous at 4 (because it isnot defined for x > 4) but is continuous from the left at 4. (c) K is continuous on the interva ls [−2, 1) and [1, 4].Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 1.1: Examples 1 through 4†When we say that one (po s s ibly infinite) interval I1is “larger” than another interval I2, we mean that I1contains I2andis not equal to I2.‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to th e chapters and sectionsof the textbook for the