Assignments Limits



AP Calculus Assignments: Limits and Continuity

|Day |Topic |Assignment |

|1 |Intro to limits |HW Limits – 1 |

|2 |Properties of limits; evaluating limits by direct substitution |HW Limits – 2 |

|3 |Evaluating limits algebraically |HW Limits – 3 |

|4 |Practice day **QUIZ** |HW Limits – 4 |

|5 |Continuity |HW Limits – 5 |

|6 |Intermediate Value Theorem |HW Limits – 6 |

|7 |Infinite limits |HW Limits – 7 |

|8 |Limits at infinity |HW Limits – 8 |

|9 |Practice day **QUIZ** |HW Limits – 9 |

|10 |Review |Review – Limits |

|11 |***TEST*** | |

AP Calculus HW: Limits – 1

1. a. Explain in your own words what is meant by the statement [pic].

b. Is it possible for the statement to be true if f(3) is undefined? Explain (or illustrate).

c. Is it possible for the statement to be true if f(3) = 10? Explain (or illustrate).

2. a. What is meant by [pic] and [pic]?

b. Is [pic] defined? Explain.

c. Is f(2) defined? Explain.

d. What happens to the function at x = 2?

3. Use the graph of f at right to evaluate the following:

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

4. Sketch a graph of a function that satisfies these conditions: [pic], [pic], [pic], [pic], f(0) is undefined and f(2) = 1.

5. Use your graphing calculator to estimate the value of the following limits. Then put the answers in your brain.

a. [pic] b. [pic] c. [pic]

6. Estimate the value of [pic].

7. One night, Kenny was doing his homework. One problem said that [pic] Kenny concluded that the function g has a root at x = 3. What happened?

AP Calculus HW: Limits – 2

1. Use the graph of f at right to evaluate the following:

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic]

2. Given that [pic], [pic], [pic] (f is continuous at x = –2), [pic], [pic] and [pic] (g is continuous at x = 16), evaluate the following limits:

a. [pic] b. [pic] c. [pic] d. [pic]

e. [pic] f. [pic] g. [pic] h. [pic]

3. Use the graphs of f and g at right to evaluate the following limits.

a. [pic] b. [pic]

c. [pic] d. [pic]

e. [pic] f. [pic]

4. If a function f is continuous at x = a (i.e., there is no “break” in the graph of f at x = a), then [pic]. Evaluating a limit in this way is called “direct substitution.” Evaluate the following limits by direct substitution:

a. [pic] b. [pic] c. [pic] d. [pic]

5. a. For[pic], try to evaluate by [pic]direct substitution. Then algebraically simplify the function and try again.

b. For[pic], try to evaluate by [pic]direct substitution. Then algebraically simplify the function and try again.

c. For[pic], try to evaluate by [pic]direct substitution. Then rationalize the denominator of the function and try again.

(This assignment is continued on the next page.)

d. For[pic], try to evaluate by [pic]direct substitution. Then algebraically simplify the function and try again.

e. If direct substitution of a limit gives the form [pic], does this automatically mean the limit DNE?

f. If direct substitution of a limit gives the form [pic], will we always be able to “fix” the function to find the limit?

6. Kenny had to evaluate [pic].

a. Kenny noticed the denominator goes to 0 and wrote “DNE.” What happened?

b. Given a second chance, Kenny noticed both numerator and denominator go to 0 so he wrote [pic].

What happened?

AP Calculus HW: Limits – 3

Evaluate the limits algebraically.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic]

8. Evaluate [pic] for f(x) = [pic]

9. If [pic],

a. Evaluate the following limits

(1) [pic] (2) [pic] (3) [pic] (4) [pic] (5) [pic]

b. Sketch the graph of f.

10. a. Evaluate [pic] b. Evaluate [pic] c. What happens to [pic] at x = a?

11. Kenny had to evaluate [pic]. Kenny got [pic]. What happened?

AP Calculus HW: Limits – 4

1. If [pic], which of the following are true?

a. [pic] b. If c is in the domain of f then f(c) = L.

c. f can be made as close as we wish to L (but not necessarily equal to L) by making x close enough to c.

2. Use the graph of f at right to evaluate the following limits.

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

3. Evaluate the following limits:

a. [pic] b. [pic] c. [pic] d. [pic]

4. Evaluate [pic] for f(x) = [pic].

5. Let f be the function [pic].

a. Evaluate f(1). b. Evaluate [pic]. c. Sketch the graph of f.

d. Explain why f is not continuous at x = 1. (Do not simply say there is a “break” in the graph there; explain why there is a break in the graph.

6. Let f be the function [pic].

7. Let f be the function [pic].

8. Make a hypothesis based on the previous three problems: What conditions must be met for a function to be continuous at some point x = c?

9. Kenny had to evaluate [pic] for f(x) = x2. He got [pic]. What happened?

AP Calculus HW: Limits – 5

1. Write the requirements for a function g to be continuous at x = k.

2. Sketch the graph of a function that has a jump discontinuity at x = –2, a removable discontinuity at x = 1 and an infinite discontinuity at x = 4.

3. The graph of f is shown at right. From the graph, name the x–values at which f is discontinuous. For each, tell what type of discontinuity it is.

4. An airport parking lot charges $2 for the first half hour or part thereof and $1 for each additional half hour or part thereof. Sketch the graph of the parking charge as a function of time parked and explain the real–life implications of the discontinuities in the graph (in other words, what do the discontinuities mean to the person who parks her car there?)

5. For each function, name the x-value(s) where the function has a discontinuity, tell what type of discontinuity it is and, if the discontinuity is removable, tell how to remove it.

a. [pic] b. [pic] c. [pic]

6. Find the value of a for which the function [pic] will be continuous at x = 2.

7. Find the values of a and b for which the function [pic] will be continuous at x = 4.

8. Kenny was given a continuous function g and told that [pic] and asked to evaluate g(5). Kenny, “learning” from a previous mistake, said there was not enough information. What happened?

AP Calculus HW: Limits – 6

1. Does the IVT apply in each case? If the theorem applies, find the guaranteed value of c. Otherwise, explain why the theorem does not apply.

a. [pic] on the interval [3, 7], N = 10.

b. [pic]on the interval [2, 10], N = 5.

c. [pic]on [0, 6], N = 4

2. The table below shows selected values of a function f that is continuous on [2, 9].

a. What is the least number of roots f may have in the interval [2, 9]? Justify your answer.

b. Would the answer be the same if f were not continuous? Explain.

3. The function f is continuous on the closed interval [–1, 1] and has values that are given in the table at right. For what values of k will the equation

f(x) = 2 have at least two solutions in the interval [–1, 1] ?

4. A function g has domain [2, 5] with g(2) = 6 and g(5) = –1. Which of the following is true?

(A) g must have a root in [2, 5].

(B) g may have a root in [2, 5].

(C) g can not have a root in [2, 5].

5. Suppose a function f is continuous on the interval [1, 5] except at x = 3 and f(1) = 2 and f(5) = 7.

Let N = 4. Draw two possible graphs of f, one that satisfies the conclusion of the IVT and another

that does not satisfy the conclusion of the IVT.

6. a. Suppose N and D are positive numbers. What happens to the value of [pic] if N is constant and D ( 0?

b. Suppose N is a positive number. Evaluate the following limits. When possible, be more specific than just DNE.

(1) [pic] (2) [pic] (3) [pic]

(4) [pic] (5) [pic] (6) [pic]

c. Based on (1) – (3) above, sketch the behavior of the graph of [pic] near x = a.

d. Based on (4) – (6) above, sketch the behavior of the graph of [pic] near x = a.

e. How would the graphs in part d change if N were a negative number?

8. Kenny graphed the function [pic]. He could see that

f(1) < 0 and f(2) > 0 so by the IVT, Kenny concluded there is a root

in (1, 2). What happened?

AP Calculus HW: Limits – 7

1. Use the graph of f at right to answer the following:

a. [pic] b. [pic]

c. [pic] d. [pic]

e. [pic]

f. Write equations for all the vertical asymptotes of f.

2. What is the difference between the statements [pic] and [pic]?

3. Can the graph of a function f intersect a vertical asymptote? Illustrate with a graph.

Evaluate the limits:

4. [pic] 5. [pic] 6. [pic] 7. [pic]

8. According to Einstein’s theory of relativity, the mass of a particle with velocity v is given by [pic] where c is the speed of light.

a. What does mo represent?

b. What happens to m as v ( c–?

9. a. Use your calculator to help evaluate each of the following limits.

(1) [pic] (2) [pic] (3) [pic]

b. Based on (1) – (3) above, hypothesize a general rule for [pic] where p and q are polynomials with leading coefficients of P and Q respectively and

(1) Degree p < degree of p (2) Degree of p = degree of q (3) Degree of p > degree of q

10. Kenny was asked to find all the asymptotes of the function [pic]. He wrote x = 1 and y = 1. What happened?

AP Calculus HW: Limits – 8

1. The graph of f at right has four asymptotes.

a. [pic] b. [pic]

c. [pic] d. [pic]

e. Write the equations of all the asymptotes of f.

2. Draw sketches to illustrate the difference between [pic] and [pic].

3. Can a function intersect a horizontal asymptote? Illustrate with a graph.

4. How many different horizontal asymptotes can one function have? Illustrate with a graph.

5. Evaluate[pic] 6. Evaluate both limits assuming a > 0: [pic] and [pic]

Find the indicated limit. Do not use your calculator.

7. [pic] 8. [pic] 9. [pic] 10. [pic]

11. [pic] 12. [pic] 13. [pic] 14. [pic]

15. [pic] 16. [pic] 17. [pic] 18. [pic]

19. [pic] 20. [pic] 21. [pic] 22. [pic]

23. Suppose P(x) and Q(x) are polynomial functions with leading coefficients 3 and 2 respectively. Evaluate [pic] if

a. The degree of P is less than the degree of Q.

b. The degree of P is equal to the degree of Q.

c. The degree of P is greater than the degree of Q.

24. A raindrop forms in the atmosphere and begins to fall to earth. If we assume air resistance is proportional to the speed of the raindrop, then the drop’s velocity as a function of time is given by [pic] where m is the mass of the raindrop, g is acceleration of gravity and k is a positive constant.

a. Find [pic]. The answer is called the “terminal velocity” of the raindrop.

b. Sketch the graph of v(t).

25. Suppose f is a function such that [pic] for all x > 5. Find [pic] and justify your answer.

26. Kenny had to evaluate [pic].

a. Kenny made the graph at right and concluded the limit is infinite.

What happened?

b. Given a second chance, Kenny noticed both numerator and

denominator go to ∞ so he wrote [pic]. What happened?

AP Calculus HW: Limits – 9

1. Evaluate the following limits:

a. [pic] b. [pic] c. [pic] d. [pic] e. [pic]

2. (No calculator.) Where does the function [pic] have a vertical asymptote?

(A) At x= –b only (B) At x = b only (C) At x = –b and x = b (D) Nowhere

3. (No calculator.) The function [pic] has how many

a. roots? b. vertical asymptotes? c. horizontal asymptotes?

4. Find the values of a and b that will make [pic] continuous for all x.

5. Suppose f is continuous on [–2, 5]. f(–2) = 6, f(3) = –4 and f(5) = –1. Which of the following are true? Justify your answers.

a. f has a root in the interval [–2, 3].

b. f has no root in the interval [3, 5].

c. [pic]

d. The equation f(x) = –2 has at least two solutions in [–2, 5].

e. [pic].

f. f has an absolute maximum value in [–2, 5].

g. f has an absolute maximum value in (–2, 5).

h. There is at least one solution to f(x) = –1 in (–2, 5).

i. There is at least one solution to f(x) = 6 in (–2, 5).

6. Evaluate [pic] for f(x) = x2 – 3x + 1.

7. Kenny was to evaluate [pic]. He reasoned [pic]. What happened?

AP Calculus Review: Limits

1. Explain what [pic] means.

2. Find the value of k such that the function [pic] is continuous at x = 2.

3. True or False:

a. [pic] b. If [pic] then [pic].

c. If [pic]for all real numbers other than x = a, and if [pic], then [pic].

d. For polynomial functions, the limits from the right and from the left at any point must exist and be equal.

e. If [pic] is continuous on the interval [0, 1], [pic] and [pic] has no roots in the interval, then [pic] on the entire interval [0, 1].

f. If [pic], then [pic] does not exist.

4. Find a rational function having a vertical asymptote at x = 3 and a horizontal asymptote at y = 2.

5. Draw a graphical counter-example to show the IVT does not hold if [pic] is not continuous in [a, b].

6. Identify all the asymptotes of the graph of the function [pic].

7. Use the graph of the function [pic] at right to evaluate the following limits.

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

(This assignment continued on the next page.)

8. Evaluate the limits without using your calculator:

a. [pic] b. [pic] c. [pic] d. [pic]

e. [pic] f. [pic] g. [pic] h. [pic]

9. Evaluate [pic]for the functions

a. f (x) = [pic] b. f(x) = x3 (From A2T: [pic])

10. Evaluate[pic] assuming a > 0 (why?).

11. Given that [pic], [pic], [pic], [pic], evaluate

a. [pic] b. [pic] c. [pic]

12. Use the graphs of the functions f and g at right to evaluate the following limits if they exist.

a. [pic] b. [pic]

c. [pic] d. [pic]

e. [pic] f. [pic]

g. [pic] h. [pic]

i. [pic]

13. Kenny didn’t worry too much about the test because he was confident he could figure out most limit problems with his calculator. What happened?

Answers to selected problems

Limits – 1

1b. Yes c. yes 2b. No c. We don’t know d. It has a “jump discontinuity.”

3a. 2 b. –1.5 c. 1.5 d. 3 e. 0 f. 2 g. 1 h. DNE i. 2

4. See graph at right for one of many possibilities.

5a. 1 b. 0 c. 1 6. e

7. Kenny died horribly.

Limits – 2

1a. 1 b. 2 c. –1 d. DNE e. –1 f. DNE g. 1

2a. –32 b. DNE c. 18 d. 4 e. 25 f. 0 g. 7* h. 3*

*Note: The answers to (g) and (h) are based on the continuity of f at x = –2 and the continuity of g at x = 16. We haven’t defined continuity yet but roughly speaking it means there is no “break” in the graph of the function. If the functions were not continuous, the limits might still be 7 and 3 but we would need more information about the functions to be sure.

3a. 3.5 b. DNE c. DNE d. –1 e. 2 f. DNE

4a. 25 b. 2/( c. 2/3 d. 0

5a. 8 b. (0.5 d. 4 d. No, it means DO MORE WORK e. No

6. Kenny died horribly. Twice in the same day.

Limits – 3

1. 3 2. –8 3. [pic] 4. –1/x2 5. 0 6. DNE 7. 4b2

8. [pic] 9a1. 0 a2. 1 a3. DNE a4. 2 a5. 4.5

10a. –2a b. 2a c. It has a “jump discontinuity” 11. Kenny died horribly.

Limits – 4

1. a and c 2a. 1 b. DNE c. 1 d. 3 e. 0 f. 0

3a. 1/2 b. 1/32 c. –2 d. (/2

4. –1/x2 5a. not defined b. 2 c. It’s the line y = x + 1 with a “hole” at (1, 2) d. f cannot be continuous at x = 1 b/c f is not even defined there.

6a. 1 b. DNE c. y = –1 for x < 1 and y = 1 for x ( 1

d. The graph has a “jump” at x = 1. This is b/c the two one-sided limits there do not have the same value.

7a. 1 b. 0 c. y = x2 – 1 except the point (1, 0) is displaced up 1 d. f(1) is not the same as the limit as x ( 1

8. f(c) must be defined; [pic] must exist and [pic] 9. Kenny died horribly.

Limits – 5

1. f(c) must be defined; [pic] must exist and [pic]

3. Removable at x = –4; infinite at x = –1; jump at x = 2; removable at x = 7; infinite at x = 10

4. See graph at right. The discontinuities indicate the price jumps up one dollar every half-hour (even if you park only one minute of that half hour).

5a. Removable at x = 3; let f(3) = 6 b. Infinite at x = 2 c. Jump at x = 2

6. –2 7. a = 4, b = 1.5 8. Come on. You should have caught on to this by now.

Limits – 6

1a. Yes; c = 5.606 b. No b/c 5 ( [0, 4] c. No b/c f has a discontinuity in [0, 6]

2. a. f has a root at x = 3. By IVT, it must have at least one root in (5, 6) and at least one more in (8, 9). Thus, minimum number of roots is three.

b. If f is not continuous, the IVT does not apply and

we can only be sure of the one root at x = 3.

3. k < 2 4. (B) 5. See graphs at right.

6a. ( b. (1) ( (2) (( (3) DNE

(4) ( (5) ( (6) (

c. d. e. Reflected over

x-axis

Limits – 7

1a. ( b. ( c. –( d. –( e. 2 f. x = –3; x = 2; x = 5

2. In the second case, we have no idea what the graph does

on the left side of the asymptote. (See graphs at right.)

3. Yes. See the graph in HW – 4#2. A function cannot

continuously pass through its vertical asymptote.

4. ( 5. (( 6. ( 7. –(

8a. mo = m(0) = rest mass, the mass when the object

is not moving b. m ( (

9a. (1) 0 (2) 2 (3) ( b. (1) 0 (2) [pic] (3) (

Limits – 8

1a. 1 b. 2 c. ( d. –( e. x = –1; x = 3; y = 1; y = 2

2. The first is a vertical asymptote at x = 3; the second is a horizontal asymptote at y = 3.

3. Yes. See the graph from #1 as x ( –(. 4. Two. See the graph from #1 again.

5. 0 6. [pic] and [pic] 7. 3 8. 0 9. (2 10. ¼

11. (1 12. (4 13. DNE 14. 0 15. 0 16. (

17. 0 18. 0 19. 1 20. ( 21. 0 22. 0

23a. 0 b. [pic] c. ( 24a. [pic]

25. We have: [pic]( [pic] ( [pic]

Limits – 9

1a. 1 /2 b. 0 c. (/2 d. 0 e. 0 2. (C) 3a. 1 b. 1 c. 2

4. a = –3/2, b = 7 5. a, c, d, e, f and h are true. 6. 2x – 3

Limits – Review

1. By making x close enough to a, we can make f(x) as close as we want to k.

2. –1/2 3. F, F, T, T, T, F 4. [pic] 6. x = 3 and y = 2

7a. 2 b. 3 c. DNE d. 0 e. 3 f. 0 g. –( h. 5 i. 1

8a. 3/7 b. DNE c. ( d. 0 e. 0 f. 0 g. e h. –∞

9a. [pic] b. 3x2 10. [pic] 11a. 9 b. DNE c. 5

12a. 3 b. 0.5 c. 3 d. 10 e. ( f. 3 g. ( h. 7 i. 0

13. When Kenny learned that it was a NO CALCULATOR test, his head exploded.

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1

a. Evaluate f(1).

b. Evaluate [pic].

c. Sketch the graph of f.

d. Explain why f is not continuous at x = 1.

a. Evaluate f(1).

b. Evaluate [pic].

c. Sketch the graph of f.

d. Explain why f is not continuous at x = 1.

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|[pic] |3 |k |5 |

x

x |2 |3 |4 |5 |6 |7 |8 |9 | |[pic] |(1 |0 |3 |1 |(2 |(5 |(3 |4 | |

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This is an example of a piecewise defined function. The domain is (((, () but the definition of the function is different on different “pieces” of the domain: linear when x < 0, part of a radical function for 0 ≤ x < 2 and part of a parabola for x ≥ 2.

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