Problem Set 70: Finding Limits: Numerical and Graphical Approaches

1. Explain the difference between a value at x=a and the limit as x approaches a.

2. Explain why we say a function does not have a limit as x approaches a if, as x approaches a, the left-hand limit is not equal to the right-hand limit.

For the following exercises, estimate the functional values and the limits from the graph of the function f provided in Figure 14.

A piecewise function with discontinuities at x = -2, x = 1, and x = 4.

Figure 14

3. limx2f(x)

4. limx2+f(x)

5. limx2f(x)

6. f(2)

7. limx1f(x)

8. limx1+f(x)

9. limx1f(x)

10. f(1)

11. limx4f(x)

12. limx4+f(x)

13. limx4f(x)

14. f(4)

For the following exercises, draw the graph of a function from the functional values and limits provided.

15. limx0f(x)=2,limx0+f(x)=3,limx2f(x)=2,f(0)=4,f(2)=1,f(3) does not exist.

16. limx2f(x)=0,limx2+=2,limx0f(x)=3,f(2)=5,f(0)

17. limx2f(x)=2,limx2+f(x)=3,limx0f(x)=5,f(0)=1,f(1)=0

18. limx3f(x)=0,limx3+f(x)=5,limx5f(x)=0,f(5)=4,f(3) does not exist.

19. limx4f(x)=6,limx6+f(x)=1,limx0f(x)=5,f(4)=6,f(2)=6

20. limx3f(x)=2,limx1+f(x)=2,limx3f(x)=4,f(3)=0,f(0)=0

21. limxπf(x)=π2,limxπf(x)=π2,limx1f(x)=0,f(π)=2,f(0) does not exist.

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as x approaches 0.

22. f(x)=(1+x)1x

23. g(x)=(1+x)2x

24. h(x)=(1+x)3x

25. i(x)=(1+x)4x

26. j(x)=(1+x)5x

27. Based on the pattern you observed in the exercises above, make a conjecture as to the limit of f(x)=(1+x)6x, g(x)=(1+x)7x, and h(x)=(1+x)nx.

For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit.

28. f(x)={|x|1,if x1x3,if x=1; a=1

29. f(x)={1x+1,if x=2(x+1)2,if x2; a=2

For the following exercises, use numerical evidence to determine whether the limit exists at x=a. If not, describe the behavior of the graph of the function near x=a. Round answers to two decimal places.

30. f(x)=x24x16x2;a=4

31. f(x)=x2x6x29;a=3

32. f(x)=x26x7x27x;a=7

33. f(x)=x21x23x+2;a=1

34. f(x)=1x2x23x+2;a=1

35. f(x)=1010x2x23x+2;a=1

36. f(x)=x6x25x6;a=32

37. f(x)=x4x2+4x+1;a=12

38. f(x)=2x4; a=4

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as x approaches the given value.

39. limx07tanx3x

40. limx4x2x4

41. limx02sinx4tanx

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit.

42. limx0ee1x

43. limx0ee1x2

44. limx0|x|x

45. limx1|x+1|x+1

46. limx5|x5|5x

47. limx11(x+1)2

48. limx11(x1)3

49. limx051e2x

50. Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: f(x)=|1xx| and g(x)=|1+xx| as x approaches 0. Use a graphing utility, if possible, to determine the left- and right-hand limits of the functions f(x) and g(x) as x approaches 0. If the functions have a limit as x approaches 0, state it. If not, discuss why there is no limit.

51. According to the Theory of Relativity, the mass m of a particle depends on its velocity v . That is

m=mo1(v2/c2)

where mo is the mass when the particle is at rest and c is the speed of light. Find the limit of the mass, m, as v approaches c.

52. Allow the speed of light, c, to be equal to 1.0. If the mass, m, is 1, what occurs to m as vc? Using the values listed in the table below, make a conjecture as to what the mass is as v approaches 1.00.

v m
0.5 1.15
0.9 2.29
0.95 3.20
0.99 7.09
0.999 22.36
0.99999 223.61