3. f(x) 4. 5. 3 2 1 10 4 -2 3 -3 6. f(x)+ f(x) 3 2 0 3.2 Continuity 159 15. k(x) = ex-1 17. r(x) = In 16. j(x) = ex 18. j(x) = In x+2 X In Exercises 19-24, (a) graph the given function, (b) find all values of x where the function is discontinuous, and (c) find the limit from the left and from the right at any values of x found in part (b). if x < 2 19. f(x) = x + 3 if 2 x 4 if x 4 7 x 1 if x < 1 20. f(x) = 0 if 1 x 4 x 2 if x 4 1 2 11 if x < -1 21. g(x) = x+2 if-1 x 3 11 if x 3 0 if x < 0 22. g(x) = x -5x if 0 x 5 2 23. h(x): = 5 if x 5 4x + 4 if x 0 x-4x+4 if x > 0 x+x-12 if x 1 24. h(x) = 3 x if x 1 In Exercises 25-28, find the value of the constant k that makes the function continuous. 3 25. f(x) = Jkx if x 2 x+k if x > 2 26. g(x) = x3+k if x 3 kx 5 if x > 3 2x 27. g(x) = x-15 x-3 kx 1 if x 3 if x 3 28. h(x) = 3x + 2x8 x + 2 3x + k if x 2 if x = -2 Find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn't exist. 7. f(x) = 5 + x x(x-2) -2x 8. f(x) = (2x + 1)(3x+6) x - 4 9. f(x) x 2 11. p(x) = x 12. q(x)=-3x3 + 2x-4x+1 -4x+11 x + 2 13. p(x) == x + 2 10. f(x) = x - 25 x + 5 29. Explain in your own words what the Intermediate Value Theorem says and why it seems plausible. 30. Explain why lim (3x + 8x) can be evaluated by substituting x = 2. x 2 In Exercises 31-32, (a) use a graphing calculator to tell where the rational function P(x)/Q(x) is discontinuous, and (b) verify your answer from part (a) by using the graphing calculator to plot Q(x) and determine where Q(x) = 0. You will need to choose the viewing window carefully. 14. r(x) 5x x-5 31. f(x) = x3 x + x + 2 0.9x24.14x - 5.4