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
- In your own words, what does it mean to "find the limit of f(x) as x approaches 3"?
- An expression of the form 00 is called _____.
- T/F: The limit of f(x) as x approaches 5 is f(5).
- Describe three situations where limx→cf(x) does not exist.
- In your own words, what is a difference quotient.
Problems
In Exercises 6-16, approximate the given limits both numerically and graphically.
6. limx→1x2+3x−5
7. limx→0x3−3x2+x−5
8. limx→0x+1x2+3x
9. limx→3x2−2x−3x2−4x+3
10. limx→−1x2+8x+7x2+6x+5
11. limx→2x2+7x+10x2−4x+4
12. limx→2, where f(x)={x+2x≤23x−5x>2.
13. limx→3, where f(x)={x2−x+1x≤32x+1x>3.
14. limx→0, where f(x)={cosxx≤0x2+3x+1x>0.
15. limx→π/2, where f(x)={sinxx≤π/2cosxx>π/2.
In Exercises 16-24, a function f and a value a are given. Approximate the limit of the difference quotient, limh→0f(a+h)−f(a)h, using h=±0.1,±0.01.
16. f(x)=−7x+2,a=3
17. f(x)=9x+0.06,a=−1
18. f(x)=x2+3x−7,a=1
19. f(x)=1x+1,a=2
20. f(x)=−4x2+5x−1,a=−3
21. f(x)=lnx,a=5
22. f(x)=sinx,a=π
23. f(x)=cosx,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→53−x+−2
6. limx→3x2−3=6
7. limx→4x2+x−5=15
8. limx→2x3−1=7
9. limx→25=5
10. limx→0e2x−1=0
11. limx→0sinx=0 (Hint: use the fact that |sinx|≤|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→cb=b.
2. Explain in your own words, without using ε−δ formality, why limx→cx=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→1f(x)=0
(b)limx→1g(x)=0
(c)limx→1f(x)/g(x)=2
What can be said about the relative sizes of f(x) and g(x) as x approaches 1?
Problems
Using:
limx→9f(x)=6limx→6f(x)=9limx→9g(x)=3limx→6g(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(3f(x)/g(x))
8. limx→9(f(x)−2g(x)g(x))
9. limx→6(f(x)3−g(x))
10. limx→9g(f(x))
11. limx→6f(g(x))
12. limx→6g(f(f(x)))
13. limx→6f(x)g(x)−f2(x)+g2(x)
Using
limx→1f(x)=2limx→10f(x)=1limx→1g(x)=0limx→10g(x)=π
evaluate the limits given in Exercises 14-17, where possible. If it is not possible to know, state so.
14. limx→1f(x)g(x)
15. limx→10cos(g(x))
16. limx→1f(x)g(x)
17. limx→1g(5f(x))
In Exercises 18-32, evaluate the given limit.
18. limx→3x2−3x+7
19. limx→π(x−3x+5)7
20. limx→π/4cosxsinx
21. limx→0lnx
22. limx→34x3−8x
23. limx→π/6cscx
24. limx→0ln(1+x)
25. limx→πx2+3x+55x2−2x−3
26.limx→π3x+11−x
27.limx→6x2−4x−12x2−13x+42
28.limx→0x2+2xx2−2x
29.limx→2x2+6x−16x2−3x+2
30.limx→2x2−5x−14x2+10x+16
31.limx→−2x2−5x−14x2+10x+16
32.limx→−1x2+9x+8x2−6x−7\
Use the Squeeze Theorem in Exercises 33-36, where appropriate, to evaluate the given limit.
33. limx→0xsin(1x)
34. limx→0sinxcos(1x2)
35. limx→1f(x), where 3x−2≤f(x)≤x3.
36. limx→3+f(x), where 6x−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→0sin3xx
38. limx→0sin5x8x
39. limx→0ln(1+x)x
40. limx→0sinxx, 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→1f(x)=5
3. T/F: If limx→1−f(x)=5, then limx→1+f(x)=5
4. T/F: If limx→1f(x)=5, then limx→1−f(x)=5
Problems
In Exercises 5-12, evaluate each expression using the given graph of f(x).
5.
6.
7.
8.
9.
10.
11.
12.
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→1f(x)
(d) f(1)
14. f(x)={2x2+5x−1x<0sinxx≥0
(a) limx→0−f(x)
(b) limx→0+f(x)
(c) limx→0f(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→−1f(x)
(d) f(−1)
(e) limx→1−f(x)
(f) limx→1+f(x)
(g) limx→1f(x)
(h) f(1)
16. f(x)={cosxx<πsinxx≥π
(a) limx→π−f(x)
(b) limx→π+f(x)
(c) limx→πf(x)
(d) f(π)
17. f(x)={1−cos2xx<asin2xx≥a, where a is a real number.
(a) limx→a−f(x)
(b) limx→a+f(x)
(c) limx→af(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→1f(x)
(d) f(1)
19. f(x)={x2x<2x+1x=2−x2+2x+4x>2
(a) limx→2−f(x)
(b) limx→2+f(x)
(c) limx→2f(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→bf(x)
(d) f(b)
21. f(x)={|x|xx≠00x=0
(a) limx→0−f(x)
(b) limx→0+f(x)
(c) limx→0f(x)
(d) f(0)
Review
22. Evaluate the limit: limx→−1x2+5x+4x2−3x−4
23. Evaluate the limit: limx→−4x2−16x2−4x−32
24. Evaluate the limit: limx→−6x2−15x+54x2−6x
25. Approximate the limit numerically: limx→0.4x2−4.4x+1.6x2−0.4x
26. Approximate the limit numerically: limx→0.2x2+5.8x−1.2x2−4.2x+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→cf(x) exists, then f is continuous at c.
6. T/F: If f is continuous at c, then limx→cf(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
12. a=1
13. a=1
14. a=0
15. a=1
16. a=4
17.
(a) a=−2
(b) a=0
(c) a=2
In Exercises 18-21, determine if f is continuous at the indicated values. If not, explain why.
18. f(x)={1x=0sinxxx>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+5x+4x2+3x+2x≠−13x=−1
(a) x=−1
(b) x=10
21. f(x)={x2−64x2−11x+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)=x2−3x+9
23. g(x)=√x2−4
24. h(k)=√1−k+√k+1
25. f(t)=√5t2−30
26. g(t)=1√1−t2
27. g(x)=11+x2
28. f(x)=ex
29. g(s)=lns
30. h(t)=cost
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+2x−4 on [1,1.5].
38. f(x)=sinx−1/2 on [0.5,0.55].
39. f(x)=ex−2 on [0.65,0.7].
40. f(x)=cosx−sinx on [0.7,0.8].
Review
41. Let f(x)={x2−5x<55xx≥5.
(a) limx→5−f(x)
(b) limx→5+f(x)
(c) limx→5f(x)
(d) f(5)
42. Numerically approximate the following limits:
(a) limx→4/5+x2−8.2x−7.2x2+5.8x+4
(b) limx→4/5−x2−8.2x−7.2x2+5.8x+4
43. Give an example of function f(x) for which limx→0f(x) does not exist.
1.6: Limits Involving Infinity
Terms and Concepts
1. T/F: If limx→5f(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→5f(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→7f(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)
10. f(x)=1(x−3)(x−5)2
(a) limx→3−f(x)
(b) limx→3+f(x)
(c) limx→3f(x)
(d) limx→5−f(x)
(e) limx→5+f(x)
(f) limx→5f(x)
11. f(x)=1ex+1
(a) limx→−∞f(x)
(b) limx→∞f(x)
(c) limx→0−f(x)
(d) limx→0+f(x)
12. f(x)=x2sin(πx)
(a) limx→−∞f(x)
(b) limx→∞f(x)
13. f(x)=cos(x)
(a) limx→−∞f(x)
(b) limx→∞f(x)
14. f(x)=2x+10
(a) limx→−∞f(x)
(b) limx→∞f(x)
In Exercises 15-18, numerically approximate the following limits:
(a) limx→3−f(x)
(b) limx→3+f(x)
(c) limx→3f(x)
15. f(x)=x2−1x2−x−6
16. f(x)=x2+5x−36x3−5x2+3x+9
17. f(x)=x2−11x+30x3−4x2−3x+18
18. f(x)=x2−9x+18x2−x−6
In Exercises 19-24, identify the horizontal and vertical asymptotes, if any, of the given function.
19. f(x)=2x2−2x−4x2+x−20
20. f(x)=−3x2−9x−65x2−10x−15
21. f(x)=x2+2−127x3−14x2−21x
22. f(x)=x2−99x−9
23. f(x)=x2−99x+27
24. f(x)=x2−1−x2−1
In Exercises 25-28, evaluate the given limit.
25. limx→∞x3+2x2+1x−5
26. limx→∞x3+2x2+15−x
27. limx→∞x3+2x2+1x2−5
28. limx→∞x3+2x2+15−x2
Review
29. Use an ε−δ proof to show that limx→15x−2=3.
30. Let limx→2f(x)=3 and limx→2g(x)=−1. Evaluate the following limits.
(a) limx→2(f+g)(x)
(b) limx→2(fg)(x)
(c) limx→2(f/g)(x)
(d) limx→2f(x)g(x)
31. Let f(x)={x2−1x<3x+5x≥3. Is f continuous everywhere?
32. Evaluate the limit: limx→clnx.
