HW 2 - Webwork questions with answers. Limit PDF

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Webwork questions with answers. Limit...

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Maryam Jafarova Assignment Homework 2 due 11/29/2020 at 11:59pm AZT

20fall-ada-math-calc1-sb

• • • • •

1. (1 point) Evaluate the following limits: 1 1. lim 2 = − x→−7 x (x + 7) 1 2. lim = x→0 x2 (x + 7) 2 = 3. lim− x − 3 x→3 2 = 4. lim x→3+ x − 3

3.099999999999 3.1 3.100000000001 No No

(score 0.4000000059604645) 3. (1 point) Evaluate the limit a3 − a a→1 a2 − 1 lim

Answer(s) submitted: • • • •

-inf inf -inf inf

Answer: Answer(s) submitted: • 1

(correct)

(correct) 2. (1 point) Below is an ”oracle” function. An oracle function is a function presented interactively. When you type in an x value, and press the –f–> button and the value f (x) appears in the right hand window. There are three lines, so you can easily calculate three different values of the function at one time. x Enter x Enter x Enter x

→ → → →

4. (1 point) Find an integer which is the limit of 1 − cos(x) ) x as x goes to 0. Enter I for infinity or DNE for does not exist. You should also try using identities to transform the expressions algebraically so that you can identify the limits without using a calculator.

f(x) result: f (x) result: f (x) result: f (x)

Note: The computer will round your inputs above to 12 places and will return the function value at the rounded xvalue. For example, entering 0.9999999999999 will result in the computer checking the function at x = 1 and returning exactly f (1) rather than a number necessarily near f (x) as x → 1− .

Answer: Answer(s) submitted: • 0

(correct) 5. (1 point) Use factoring to calculate the following limit.

Determine the limits for the function f (x) as x approaches 3.1. lim f (x) =

x→3.1−

f (3.1) =

lim

y 4 − a4

y→a y3 − a3

lim f (x) =

Answer: Hint: Try doing this numerically for a couple of values of y and a.

x→3.1+

Are all of these values the same? ? . If so then the function is continuous at 3.1 Are the left and right limits the same at 3.1? ? . If so then this function is almost continuous and could be made continuous by redefining one value of the function namely f (3.1).

Answer(s) submitted: • 4a/3

(correct)

Answer(s) submitted: 1

6. (1 point) Evaluate

9. (1 point) Evaluate the limit

lim (x − 2)5 (4x2 )

x2 + 14x + 40 x→10 x + 10

x→2

lim

Answer: Answer: Use the space below to enter the letters corresponding to the Limit Laws that you used to find this limit:

Answer(s) submitted: • 14

Limit Laws A. B. C. D. E. F. G.

(correct)

Product Law Sum Law Constant Multiple Law Difference Law Quotient Law Root Law Power Law

10. (1 point) Evaluate the limit x2 + 11x + 18 x+9 x→−9 lim

Answer:

Answer:

Answer(s) submitted:

Answer(s) submitted: • 0 • ACDG

• -7

(correct)

(correct)

7. (1 point) Suppose

11. (1 point) The slope of the tangent line to the graph of y = 5x3 at the point (2, 40) is

6x − 22 ≤ f (x) ≤ x2 − 13

5x3 − 40 . x→2 x − 2 lim

Use this to compute the following limit.

By trying values of x near 2, estimate the slope of the tangent line.

lim f (x) x→3

Answer:

The slope of the tangent line is (roughly)

.

Answer(s) submitted: • 60

What theorem did you use to arrive at your answer? Answer:

(correct)

Answer(s) submitted: • -4 • squeeze

12. (1 point) 7 98 Let f (y) = y−7 − y2 −49 Calculate lim f (y) by first finding a continuous function which

(correct)

8. (1 point) Evaluate the limit lim Answer:

y→7

is equal to f everywhere except y = 7. lim f (y) =

x−3

x→3 x2 + 7x − 30

y→7

Answer(s) submitted: • 1/2

Answer(s) submitted: • 1/13

(correct)

(correct)

2

13. (1 point) Let f be the function below. You can click on the graph to get a larger image.

f (x)

g(x)

The graphs of f and g are given above. You may click on the graphs to get larger images of them. Use the graphs to evaluate each quantity below. Write DNE if the limit or value does not exist (or if it’s infinity). 1. lim [ f (x)/g(x)] x→3−

2. f (2)g(2) 3. lim− [ f (x)g(x)] x→2

4. lim− [ f (x)g(x)] x→3

Answer(s) submitted: • • • •

Determine the limits for the function f at 1. Note that the limits are all integers (e.g. · · · , −2, −1, 0, 1, 2, · · ·).

(correct)

lim f (x) =

15. (1 point) ( 5 − x+4 , if x < −4 Let f (x) = 2x + 8, if x > −4 Calculate the following limits. Enter DNE if the limit does not exist.

x→1−

f (1) = lim f (x) =

x→1+

Is this function continuous at 1? (yes/no):

lim f (x) =

x→−4−

lim f (x) =

Can one change the value of this function at 1 to some value other than its current value, and make the function continuous at 1? (yes/no):

x→−4+

lim f (x) =

x→−4

Answer(s) submitted:

Answer(s) submitted: • • • • •

2 3 9 2

• DNE • 0 • DNE

1 0 0 no no

(correct) 16. (1 point) Evaluate the limits. If a limit does not exist, enter DNE. |x + 6| = lim x→−6+ x + 6 |x + 6| = lim − x→−6 x + 6 |x + 6| = lim x→−6 x + 6

(correct)

14. (1 point)

Answer(s) submitted: 3

• 1

20. f (3)g(3) 21. lim [ f (g(x))]

• -1 • DNE

x→3−

(correct)

22. f (3)/g(3) 23. lim [ f (x)/g(x)]

17. (1 point) 49−a √ . Let f (a) = 7− a Calculate lim f (a) by first finding a continuous function which

x→3+

24. f (g(1)) Answer(s) submitted:

a→49

is equal to f everywhere except a = 49. lim f (a) =

• • • • • • • • • • • • • • • • • • • • • • • •

a→49

Answer(s) submitted: • 14

(correct) 18. (1 point)

f (x)

g(x)

The graphs of f (x) and g(x) are given above. Use them to evaluate each quantity below. Write DNE if the limit or value does not exist (or if it’s infinity). 1. lim [ f (x) + g(x)]

1 3 DNE 1 3 DNE 1/2 0 1 4 3 0 DNE 0 0 0 3 2 3 0 3 DNE DNE 3

x→1+

(correct)

x→3−

19. (1 point) Calculate:

2. f (g(3)) 3. lim [ f (x)/g(x)] 4. f (1) + g(1) 5. lim+ [ f (x) + g(x)] x→3

lim sin 5x x→0+ |5x|

6. lim+ [ f (x)/g(x)] x→1

7. lim [ f (x)/g(x)]

and lim

x→1−

8. lim [ f (g(x))]

x→0−

x→1−

sin 5x |5x|

=

=

Answer(s) submitted:

9. lim [ f (x) + g(x)] x→3−

• 1 • -1

10. f (3) + g(3) 11. lim− [ f (x) + g(x)] x→1

(correct)

12. lim+ [ f (x)g(x)] x→1

13. f (1)/g(1) 14. lim [ f (x)g(x)]

20. (1 point) Evaluate the limit: 5h lim 1−cos = h

x→3−

15. f (1)g(1) 16. lim [ f (x)g(x)]

h→ 2π

x→3+

Answer(s) submitted:

17. lim [ f (g(x))]

• 2/pi

x→3+

18. lim [ f (x)g(x)]

(correct)

x→1−

19. lim+ [ f (g(x))] x→1

4

26. (1 point) Evaluate the limit: lim sin6h 6h =

21. (1 point)

h→0

Answer(s) submitted: • 1

(correct) 27. (1 point) Evaluate the limit sin 3h = h→0 5h Enter DNE if the limit does not exist. Limit = lim

In the figure, is f (x) squeezed by u(x) and l(x) Answer(s) submitted: • 3/5

? 1. at x = 4?

(correct)

? 2. at x = 3?

28. (1 point) Evaluate the limit: (6h) lim 1−cos = h

Answer(s) submitted: • F • T

h→0

Answer(s) submitted:

(correct)

• 0

22. (1 point) Evaluate the limit: sin2 (2t) = lim t t→0

(correct) 29. (1 point) Evaluate the limit: lim sin(z/3) sinz =

Answer(s) submitted:

z→0

• 0

Answer(s) submitted:

(correct)

• 1/3

(correct)

23. (1 point) Evaluate the limit: x2 = lim 2 x→0 sin (6x)

30. (1 point) Evaluate the limit: x sin 4x lim sin9 sin4x sin 6x = x→0

Answer(s) submitted:

Answer(s) submitted:

• 1/36

• 3/2

(correct)

(correct)

24. (1 point) Evaluate the limit: lim tan8x5x =

31. (1 point) Evaluate the limit: sint = lim π t→ 4 t

x→0

Answer(s) submitted:

Answer(s) submitted:

• 5/8

• 2ˆ(3/2)/pi

(correct)

(correct)

25. (1 point) Evaluate the limit: 2t lim tan t sect =

32. (1 point) Evaluate the limit: lim tan3x tan8x =

t→0

x→0

Answer(s) submitted:

Answer(s) submitted:

• 2

• 0.375

(correct)

(correct) 5

39. (1 point) In this problem we consider three functions f . Each of them is continuous at x = 0, i.e.,

33. (1 point) Evaluate the limit: 8t lim 1−cos sin6t = t→0

lim f (x) = f (0).

Answer(s) submitted:

x−→0

• 0

In order to show by the ε/δ definition that this is true one has to give a definition of δ in terms of ε such that

(correct)

|x − 0| < δ

34. (1 point) Use the Squeeze Theorem to evaluate the limit: lim x cos(8/x) =

=⇒

| f (x) − f (0)| < ε.

Match these choices of δ

x→0

Answer(s) submitted:

1. δ = ε2 2. δ = ε√ 3. δ = ε with the functions so that that choice of δ establishes continuity of the function (at x = 0). You can use each choice only once. Enter the reference numbers of the given functions in the appropriate answer boxes.

• 0

(correct) 35. (1 point) Use the √ Squeeze Theorem to evaluate the limit: lim 3x · ecos(2π/x) = x→0+

Answer(s) submitted:

f (x) = x: f (x) = x√2 : f (x) = x:

• 0

(correct)

Answer(s) submitted:

36. (1 point) Find the largest number δ such that   1 1   − if 0 < |x − 1.6| < δ then  x2 1.62  < 1.3 .

• 2 • 3 • 1

(correct) √ 40. (1 point) Use the given graph of f (x) = x to find the largest number δ such that √   x − 3 < 0.2 . if 0 < |x − 9| < δ then

δ=

Answer(s) submitted: • 1.6-(1/sqrt(1/(1.6ˆ2)+1.3))

(correct) 37. (1 point) Let f (x) = −3x + 4. Find the largest δ so that | f (x) − f (a)| < ε when |x − a| < δ. δ= Note: Type the word epsilon for ε in your answer. Answer(s) submitted: • epsilon/3

δ= Answer(s) submitted:

(correct)

• 9-2.8ˆ2

38. (1 point) Let p(x) = 4 + 4x + 5x2 . Find the largest δ so that |p(x) − p(3.2)| < 0.00019 when |x − 3.2| < δ. δ=

(correct)

41. (1 point) ( x2 If f (x) = 0 find lim f (x).

Answer(s) submitted: • 0.0001

if x is rational if x is irrational

x→0

Answer(s) submitted:

(incorrect) 6

• 0

(correct)

Answer(s) submitted: • A • A • A

42. (1 point) Which of the following is the meaning of the statement lim f (x) = ∞? x→3

(correct)

(a) As x approaches 3 from the right, f (x) is getting arbitrarily large. (b) As x approaches 3 from the left, f (x) is getting arbitrarily large. (c) As x approaches 3 from either side, f (x) is getting arbitrarily large. (d) As x approaches 3 from either side, who knows where f (x) is going.

45. (1 point) Suppose lim its. (a) lim f (x) = x→3 √ x f (x) = (b) lim 2 x→3 x − 5

(correct)

46. (1 point) Order 7 of the following sentences so that they form a logical direct proof of the statement:

x→3

as follows. Given any ε > 0, we need to find a number δ > 0 such that if 0 < |x − 3| < δ, then |(x2 − 6x) − (−9)| < ε. What is the (largest) choice of δ that is certain to work? (Your answer will involve ε. When entering your answer, type e in place of ε.)

lim (x2 ) = 16. x→4

δ=

• δ < 2 =⇒ 0 < x+4 < 8+δ < 8+2 • Use Wolfram-Alpha to figure this out. • Assume |(x2 ) − 16| ≤ ε. • Suppose ε > 0 • |x − 4||x + 4| < 10ε (10) = ε • Assume |x − 4| < δ and |(x2 ) − 16| ≤ ε • There is no limit. • Therefore, lim(x2 ) = 16.

Answer(s) submitted: • sqrt(e)

(correct)

If lim f (x) = L, then f (a) = L x→a

• A. False • B. True If lim f (x) = L, and limg(x) = L then f (a) = g(a) x→a

x→a

.

(correct)

43. (1 point) Let f (x) = x2 − 6x. To prove that lim f (x) = −9, we proceed

The limit lim

.

• 9 • (sqrt(3)*9)/4

• c

• A. False • B. True

f (x) = 1. Find the following limx2

Answer(s) submitted:

Answer(s) submitted:

x→a

x→3

• • • •

f (x) does not exist if g(a) = 0 g(x)

x→4

Then |x − 4| < δ Choose δ > 0 so that δ < 10ε and δ < 2. 8−δ < x+4 < 8+δ Assume 0 < |x − 4| < δ

Answer(s) submitted:

• P1-3,P1-9,P1-11,P1-10,P1-0,P1-4,P1-7

• A. False • B. True

(correct)

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