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.

Figure 14
3. limx→−2−f(x)
4. limx→−2+f(x)
5. limx→−2f(x)
6. f(−2)
7. limx→−1−f(x)
8. limx→1+f(x)
9. limx→1f(x)
10. f(1)
11. limx→4−f(x)
12. limx→4+f(x)
13. limx→4f(x)
14. f(4)
For the following exercises, draw the graph of a function from the functional values and limits provided.
15. limx→0−f(x)=2,limx→0+f(x)=−3,limx→2f(x)=2,f(0)=4,f(2)=−1,f(−3) does not exist.
16. limx→2−f(x)=0,limx→2+=−2,limx→0f(x)=3,f(2)=5,f(0)
17. limx→2−f(x)=2,limx→2+f(x)=−3,limx→0f(x)=5,f(0)=1,f(1)=0
18. limx→3−f(x)=0,limx→3+f(x)=5,limx→5f(x)=0,f(5)=4,f(3) does not exist.
19. limx→4f(x)=6,limx→6+f(x)=−1,limx→0f(x)=5,f(4)=6,f(2)=6
20. limx→−3f(x)=2,limx→1+f(x)=−2,limx→3f(x)=−4,f(−3)=0,f(0)=0
21. limx→πf(x)=π2,limx→−πf(x)=π2,limx→1−f(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 x≠1x3,if x=1; a=1
29. f(x)={1x+1,if x=−2(x+1)2,if x≠−2; 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)=x2−4x16−x2;a=4
31. f(x)=x2−x−6x2−9;a=3
32. f(x)=x2−6x−7x2−7x;a=7
33. f(x)=x2−1x2−3x+2;a=1
34. f(x)=1−x2x2−3x+2;a=1
35. f(x)=10−10x2x2−3x+2;a=1
36. f(x)=x6x2−5x−6;a=32
37. f(x)=x4x2+4x+1;a=−12
38. f(x)=2x−4; 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. limx→07tanx3x
40. limx→4x2x−4
41. limx→02sinx4tanx
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. limx→0ee1x
43. limx→0ee−1x2
44. limx→0|x|x
45. limx→−1|x+1|x+1
46. limx→5|x−5|5−x
47. limx→−11(x+1)2
48. limx→11(x−1)3
49. limx→051−e2x
50. Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: f(x)=|1−xx| 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=mo√1−(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 v→c? 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 |
Candela Citations
- Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution. License Terms: Download for free at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface