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1.E: Applications of Limits (Exercises)

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    Dec 21, 2020
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1.1: An Introduction to Limits

Terms and Concepts

  1. In your own words, what does it mean to "find the limit of f⁡(x) as x approaches 3"?
  2. An expression of the form 00 is called _____.
  3. T/F: The limit of f⁡(x) as x approaches 5 is f⁡(5).
  4. Describe three situations where limx→c⁡f⁡(x) does not exist.
  5. In your own words, what is a difference quotient.

Problems

In Exercises 6-16, approximate the given limits both numerically and graphically.

6. limx→1⁡x2+3⁢x−5

7. limx→0⁡x3−3⁢x2+x−5

8. limx→0⁡x+1x2+3⁢x

9. limx→3⁡x2−2⁢x−3x2−4⁢x+3

10. limx→−1⁡x2+8⁢x+7x2+6⁢x+5

11. limx→2⁡x2+7⁢x+10x2−4⁢x+4

12. limx→2, where f⁡(x)={x+2x≤23⁢x−5x>2.

13. limx→3, where f⁡(x)={x2−x+1x≤32⁢x+1x>3.

14. limx→0, where f⁡(x)={cos⁡xx≤0x2+3⁢x+1x>0.

15. limx→π/2, where f⁡(x)={sin⁡xx≤π/2cos⁡xx>π/2.

In Exercises 16-24, a function f and a value a are given. Approximate the limit of the difference quotient, limh→0⁡f⁡(a+h)−f⁡(a)h, using h=±0.1,±0.01.

16. f⁡(x)=−7⁢x+2,a=3

17. f⁡(x)=9⁢x+0.06,a=−1

18. f⁡(x)=x2+3⁢x−7,a=1

19. f⁡(x)=1x+1,a=2

20. f⁡(x)=−4⁢x2+5⁢x−1,a=−3

21. f⁡(x)=ln⁡x,a=5

22. f⁡(x)=sin⁡x,a=π

23. f⁡(x)=cos⁡x,a=π

1.2: Epsilon-Delta Definition of a Limit

Terms and Concepts

1. What is wrong with the following "definition" of a limit?

"The limit of f⁡(x), as x approaches a, is K″ means that given any δ>0 there exists ϵ>0 such that whenever |f⁡(x)−K|<ϵ, we have |x−a|<δ.

2. Which is given first in establishing a limit, the x-tolerance or the y-tolerance?

3. T/F: ϵ must always be positive.

4. T/F: δ must always be positive.

Problems

In Exercises 5-11, prove the given limit using an ϵ−δ proof.

5. limx→5⁡3−x+−2

6. limx→3⁡x2−3=6

7. limx→4⁡x2+x−5=15

8. limx→2⁡x3−1=7

9. limx→2⁡5=5

10. limx→0⁡e2⁢x−1=0

11. limx→0⁡sin⁡x=0 (Hint: use the fact that |sin⁡x|≤|x|, with equality only when x=0.)

1.3: Finding Limits Analytically

Terms and Concepts

1. Explain in your own words, without using ε−δ formality, why limx→c⁡b=b.

2. Explain in your own words, without using ε−δ formality, why limx→c⁡x=c.

3. What does the text mean when it says that certain functions’ “behavior is ‘nice’ in terms of limits”? What, in particular, is “nice”?

4. Sketch a graph that visually demonstrates the Squeeze Theorem.

5. You are given the following information:

(a) limx→1⁡f⁡(x)=0

(b) limx→1⁡g⁡(x)=0

(c) limx→1⁡f⁡(x)/g⁡(x)=2

What can be said about the relative sizes of f⁡(x) and g⁡(x) as x approaches 1?

Problems

Using:

(1.E.1)limx→9⁡f⁡(x)=6limx→6⁡f⁡(x)=9(1.E.2)limx→9⁡g⁡(x)=3limx→6⁡g⁡(x)=3

evaluate the limits given in Exercises 6-13, where possible. If it is not possible to know, state so.

6. limx→9⁡(f⁡(x)+g⁡(x))

7. limx→9⁡(3⁢f⁡(x)/g⁡(x))

8. limx→9⁡(f⁡(x)−2⁢g⁡(x)g⁡(x))

9. limx→6⁡(f⁡(x)3−g⁡(x))

10. limx→9⁡g⁡(f⁡(x))

11. limx→6⁡f⁡(g⁡(x))

12. limx→6⁡g⁡(f⁡(f⁡(x)))

13. limx→6⁡f⁡(x)⁢g⁡(x)−f2⁡(x)+g2⁡(x)

Using

(1.E.3)limx→1⁡f⁡(x)=2limx→10⁡f⁡(x)=1(1.E.4)limx→1⁡g⁡(x)=0limx→10⁡g⁡(x)=π

evaluate the limits given in Exercises 14-17, where possible. If it is not possible to know, state so.

14. limx→1⁡f⁡(x)g⁡(x)

15. limx→10⁡cos⁡(g⁡(x))

16. limx→1⁡f⁡(x)⁢g⁡(x)

17. limx→1⁡g⁡(5⁢f⁡(x))

In Exercises 18-32, evaluate the given limit.

18. limx→3⁡x2−3⁢x+7

19. limx→π⁡(x−3x+5)7

20. limx→π/4⁡cos⁡x⁢sin⁡x

21. limx→0⁡ln⁡x

22. limx→3⁡4x3−8⁢x

23. limx→π/6⁡csc⁡x

24. limx→0⁡ln⁡(1+x)

25. limx→π⁡x2+3⁢x+55⁢x2−2⁢x−3

26.limx→π⁡3⁢x+11−x

27.limx→6⁡x2−4⁢x−12x2−13⁢x+42

28.limx→0⁡x2+2⁢xx2−2⁢x

29.limx→2⁡x2+6⁢x−16x2−3⁢x+2

30.limx→2⁡x2−5⁢x−14x2+10⁢x+16

31.limx→−2⁡x2−5⁢x−14x2+10⁢x+16

32.limx→−1⁡x2+9⁢x+8x2−6⁢x−7\

Use the Squeeze Theorem in Exercises 33-36, where appropriate, to evaluate the given limit.

33. limx→0⁡x⁢sin⁡(1x)

34. limx→0⁡sin⁡x⁢cos⁡(1x2)

35. limx→1⁡f⁡(x), where 3⁢x−2≤f⁡(x)≤x3.

36. limx→3+⁡f⁡(x), where 6⁢x−9≤f⁡(x)≤x2 on [0,3].

Exercises 37-40, challenge your understanding of limits but can be evaluated using the knowledge gained in this section.

37. limx→0⁡sin⁡3⁢xx

38. limx→0⁡sin⁡5⁢x8⁢x

39. limx→0⁡ln⁡(1+x)x

40. limx→0⁡sin⁡xx, where x is measured in degrees not radians.

1.4: One Sided Limits

Terms and Concepts

1. What are the three ways in which a limit may fail to exist?

2. T/F: If limx→1−⁡f⁡(x)=5, then limx→1⁡f⁡(x)=5

3. T/F: If limx→1−⁡f⁡(x)=5, then limx→1+⁡f⁡(x)=5

4. T/F: If limx→1⁡f⁡(x)=5, then limx→1−⁡f⁡(x)=5

Problems

In Exercises 5-12, evaluate each expression using the given graph of f⁡(x).

5.
clipboard_e4f5a8bce92d31a43cde475d21c7bbb8f.png

6.
146.PNG

7.
147.PNG

8.
148.PNG

9.
149.PNG

10.
1410.PNG

11.
1411.PNG

12.
1412.PNG

In Exercises 13-21, evaluate the given limits of the piecewise defined functions f.

13. f⁡(x)={x+1x≤1x2−5x>1
(a) limx→1−⁡f⁡(x)
(b) limx→0+⁡f⁡(x)
(c) limx→1⁡f⁡(x)
(d) f⁡(1)

14. f⁡(x)={2⁢x2+5⁢x−1x<0sin⁡xx≥0
(a) limx→0−⁡f⁡(x)
(b) limx→0+⁡f⁡(x)
(c) limx→0⁡f⁡(x)
(d) f⁡(0)

15. f⁡(x)={x−⁢1x<−1x3+1−1≤x≤1x2+1x>1
(a) limx→−1−⁡f⁡(x)
(b) limx→1+⁡f⁡(x)
(c) limx→−1⁡f⁡(x)
(d) f⁡(−1)
(e) limx→1−⁡f⁡(x)
(f) limx→1+⁡f⁡(x)
(g) limx→1⁡f⁡(x)
(h) f⁡(1)

16. f⁡(x)={cos⁡xx<πsin⁡xx≥π
(a) limx→π−⁡f⁡(x)
(b) limx→π+⁡f⁡(x)
(c) limx→π⁡f⁡(x)
(d) f⁡(π)

17. f⁡(x)={1−cos2⁡xx<asin2⁡xx≥a, where a is a real number.
(a) limx→a−⁡f⁡(x)
(b) limx→a+⁡f⁡(x)
(c) limx→a⁡f⁡(x)
(d) f⁡(a)

18. f⁡(x)={x+1x<11x=1x−1x>1
(a) limx→1−⁡f⁡(x)
(b) limx→1+⁡f⁡(x)
(c) limx→1⁡f⁡(x)
(d) f⁡(1)

19. f⁡(x)={x2x<2x+1x=2−x2+2⁢x+4x>2
(a) limx→2−⁡f⁡(x)
(b) limx→2+⁡f⁡(x)
(c) limx→2⁡f⁡(x)
(d) f⁡(2)

20. f⁡(x)={a⁢(x−b)2+cx<ba⁢(x−b)+cx≥b, where a, b and c are real numbers.
(a) limx→b−⁡f⁡(x)
(b) limx→b+⁡f⁡(x)
(c) limx→b⁡f⁡(x)
(d) f⁡(b)

21. f⁡(x)={|x|xx≠00x=0
(a) limx→0−⁡f⁡(x)
(b) limx→0+⁡f⁡(x)
(c) limx→0⁡f⁡(x)
(d) f⁡(0)

Review

22. Evaluate the limit: limx→−1⁡x2+5⁢x+4x2−3⁢x−4

23. Evaluate the limit: limx→−4⁡x2−16x2−4⁢x−32

24. Evaluate the limit: limx→−6⁡x2−15⁢x+54x2−6⁢x

25. Approximate the limit numerically: limx→0.4⁡x2−4.4⁢x+1.6x2−0.4⁢x

26. Approximate the limit numerically: limx→0.2⁡x2+5.8⁢x−1.2x2−4.2⁢x+0.8

1.5: Continuity

Terms and Concepts

1. In your own words, describe what it means for a function to be continuous.

2. In your own words, describe what the Intermediate Value Theorem states.

3. What is a “root” of a function?

4. Given functions f and g on an interval I, how can the Bisection Method be used to find a value c where f⁡(c)=g⁡(c)?

5. T/F: If f is defined on an open interval containing c, and limx→c⁡f⁡(x) exists, then f is continuous at c.

6. T/F: If f is continuous at c, then limx→c⁡f⁡(x) exists

7. T/F: If f is continuous at c, then limx→c+⁡f⁡(x)=f⁡(c).

8. T/F: If f is continuous on [a, b], then limx→a−⁡f⁡(x)=f⁡(a).

9. T/F: If f is continuous on [0, 1) and [1, 2), then f is continuous on [0, 2).

10. T/F: The sum of continuous functions is also continuous.

Problems

In Exercises 11-17, a graph of a function f is given along with a value a. Determine if f is continuous at a; if it is not, state why it is not.

11. a=1
1511.PNG

12. a=1
1512.PNG

13. a=1
1513.PNG

14. a=0
1514.PNG

15. a=1
1515.PNG

16. a=4
1516.PNG

17.
(a) a=−2
(b) a=0
(c) a=2
1517.PNG

In Exercises 18-21, determine if f is continuous at the indicated values. If not, explain why.

18. f⁡(x)={1x=0sin⁡xxx>0
(a) x=0
(b) x=π

19. f⁡(x)={x3−xx<1x−2x≥1
(a) x=0
(b) x=1

20. f⁡(x)={x2+5⁢x+4x2+3⁢x+2x≠−13x=−1
(a) x=−1
(b) x=10

21. f⁡(x)={x2−64x2−11⁢x+24x≠85x=8
(a) x=0
(b) x=8

In Exercises 22-32, give the intervals on which the given function is continuous.

22. f⁡(x)=x2−3⁢x+9

23. g⁡(x)=x2−4

24. h⁡(k)=1−k+k+1

25. f⁡(t)=5⁢t2−30

26. g⁡(t)=11−t2

27. g⁡(x)=11+x2

28. f⁡(x)=ex

29. g⁡(s)=ln⁡s

30. h⁡(t)=cos⁡t

31. f⁡(k)=1−ek

32. f⁡(x)=sin⁡(ex+x2)

33. Let f be continuous on [1,5] where f⁡(1)=−2 and f⁡(5)=−10. Does a value 1<c<5 exist such that f⁡(c)=−9? Why/why not?

34. Let g be continuous on [-3,7] where g⁡(0)=0 and g⁡(2)=25. Does a value −3<c<7 exist such that g⁡(c)=15? Why/why not?

35. Let f be continuous on [-1,1] where f⁡(−1)=−10 and f⁡(1)=10. Does a value −1<c<1 exist such that f⁡(c)=11? Why/why not?

36. Let h be continuous on [-1,1] where h⁡(−1)=−10 and h⁡(1)=10. Does a value −1<c<1 exist such that h⁡(c)=0? Why/why not?

In Exercises 37-40, use the Bisection Method to approximate, accurate to two decimal places, the value of the root of the given function in the given interval.

37. f⁡(x)=x2+2⁢x−4 on [1,1.5].

38. f⁡(x)=sin⁡x−1/2 on [0.5,0.55].

39. f⁡(x)=ex−2 on [0.65,0.7].

40. f⁡(x)=cos⁡x−sin⁡x on [0.7,0.8].

Review

41. Let f⁡(x)={x2−5x<55⁢xx≥5.
(a) limx→5−⁡f⁡(x)
(b) limx→5+⁡f⁡(x)
(c) limx→5⁡f⁡(x)
(d) f⁡(5)

42. Numerically approximate the following limits:
(a) limx→4/5+⁡x2−8.2⁢x−7.2x2+5.8⁢x+4
(b) limx→4/5−⁡x2−8.2⁢x−7.2x2+5.8⁢x+4

43. Give an example of function f⁡(x) for which limx→0⁡f⁡(x) does not exist.

1.6: Limits Involving Infinity

Terms and Concepts

1. T/F: If limx→5⁡f⁡(x)=∞, then we are implicitly stating that the limit exists.

2. T/F: If limx→∞⁡f⁡(x)=5, then we are implicitly stating that the limit exists.

3. T/F: If limx→1−⁡f⁡(x)=−∞, then limx→1+⁡f⁡(x)=∞.

4. T/F: If limx→5⁡f⁡(x)=∞, then f has a vertical asymptote at x=5.

5. T/F: ∞/0 is not an indeterminate form.

6. List 5 indeterminate forms.

7. Construct a function with a vertical asymptote at x = 5 and a horizontal asymptote at y = 5.

8. Let limx→7⁡f⁡(x)=∞. Explain how we know that f is/is not continuous at x=7.

Problems

In Exercises 9-14, evaluate the given limits using the graph of the function.

9. f⁡(x)=1(x+1)2
(a) limx→−1−⁡f⁡(x)
(b) limx→−1+⁡f⁡(x)
169.PNG

10. f⁡(x)=1(x−3)⁢(x−5)2
(a) limx→3−⁡f⁡(x)
(b) limx→3+⁡f⁡(x)
(c) limx→3⁡f⁡(x)
(d) limx→5−⁡f⁡(x)
(e) limx→5+⁡f⁡(x)
(f) limx→5⁡f⁡(x)
1610.PNG

11. f⁡(x)=1ex+1
(a) limx→−∞⁡f⁡(x)
(b) limx→∞⁡f⁡(x)
(c) limx→0−⁡f⁡(x)
(d) limx→0+⁡f⁡(x)
1611.PNG

12. f⁡(x)=x2⁢sin⁡(π⁢x)
(a) limx→−∞⁡f⁡(x)
(b) limx→∞⁡f⁡(x)
1612.PNG

13. f⁡(x)=cos⁡(x)
(a) limx→−∞⁡f⁡(x)
(b) limx→∞⁡f⁡(x)
1613.PNG

14. f⁡(x)=2x+10
(a) limx→−∞⁡f⁡(x)
(b) limx→∞⁡f⁡(x)
1614.PNG

In Exercises 15-18, numerically approximate the following limits:
(a) limx→3−⁡f⁡(x)
(b) limx→3+⁡f⁡(x)
(c) limx→3⁡f⁡(x)

15. f⁡(x)=x2−1x2−x−6

16. f⁡(x)=x2+5⁢x−36x3−5⁢x2+3⁢x+9

17. f⁡(x)=x2−11⁢x+30x3−4⁢x2−3⁢x+18

18. f⁡(x)=x2−9⁢x+18x2−x−6

In Exercises 19-24, identify the horizontal and vertical asymptotes, if any, of the given function.

19. f⁡(x)=2⁢x2−2⁢x−4x2+x−20

20. f⁡(x)=−3⁢x2−9⁢x−65⁢x2−10⁢x−15

21. f⁡(x)=x2+2−127⁢x3−14⁢x2−21⁢x

22. f⁡(x)=x2−99⁢x−9

23. f⁡(x)=x2−99⁢x+27

24. f⁡(x)=x2−1−x2−1

In Exercises 25-28, evaluate the given limit.

25. limx→∞⁡x3+2⁢x2+1x−5

26. limx→∞⁡x3+2⁢x2+15−x

27. limx→∞⁡x3+2⁢x2+1x2−5

28. limx→∞⁡x3+2⁢x2+15−x2

Review

29. Use an ε−δ proof to show that limx→1⁡5⁢x−2=3.

30. Let limx→2⁡f⁡(x)=3 and limx→2⁡g⁡(x)=−1. Evaluate the following limits.
(a) limx→2⁡(f+g)⁡(x)
(b) limx→2⁡(f⁡g)⁢(x)
(c) limx→2⁡(f⁡/g)⁢(x)
(d) limx→2⁡f⁡(x)g⁡(x)

31. Let f⁡(x)={x2−1x<3x+5x≥3. Is f continuous everywhere?

32. Evaluate the limit: limx→c⁡ln⁡x.

1.E: Applications of Limits (Exercises) is shared under a CC BY-NC 3.0 license and was authored, remixed, and/or curated by LibreTexts.

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