Limits Assignments

CALCULUS I Assignment Problems

Limits

Paul Dawkins

Calculus I

© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

Table of Contents Preface ............................................................................................................................................. 1 Limits .............................................................................................................................................. 1

Introduction .............................................................................................................................................. 1 Rates of Change and Tangent Lines ......................................................................................................... 2 The Limit ................................................................................................................................................... 5 One-Sided Limits....................................................................................................................................... 8 Limit Properties ...................................................................................................................................... 10 Computing Limits ................................................................................................................................... 12 Infinite Limits .......................................................................................................................................... 15 Limits At Infinity, Part I .......................................................................................................................... 17 Limits At Infinity, Part II......................................................................................................................... 18 Continuity ................................................................................................................................................ 20 The Definition of the Limit ..................................................................................................................... 24

Preface Here are a set of problems for my Calculus I notes. These problems do not have any solutions available on this site. These are intended mostly for instructors who might want a set of problems to assign for turning in. I try to put up both practice problems (with solutions available) and these problems at the same time so that both will be available to anyone who wishes to use them. As with the set of practice problems I write these as I get the time and some sections will have only a few problems at this point and others won’t have any problems in them yet. Rest assured that I’m always trying to get more problems written but this site has been written and maintained in my spare time and so I usually cannot devote as much time as I’d like to the site.

Limits

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems.

http://tutorial.math.lamar.edu/terms.aspx

Calculus I

© 2007 Paul Dawkins 2 http://tutorial.math.lamar.edu/terms.aspx

Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Tangent Lines and Rates of Change The Limit One-Sided Limits Limit Properties Computing Limits Infinite Limits Limits At Infinity, Part I Limits At Infinity, Part II Continuity The Definition of the Limit – No problems have been written for this section yet.

Rates of Change and Tangent Lines 1. For the function ( ) 3 23f x x x= − and the point P given by 3x = answer each of the

following questions.

(a) For the points Q given by the following values of x compute (accurate to at least 8 decimal places) the slope, PQm , of the secant line through points P and Q.

(i) 3.5 (ii) 3.1 (iii) 3.01 (iv) 3.001 (v) 3.0001 (vi) 2.5 (vii) 2.9 (viii) 2.99 (ix) 2.999 (x) 2.9999

(b) Use the information from (a) to estimate the slope of the tangent line to ( )f x at 3x =

and write down the equation of the tangent line.

2. For the function ( ) 2 4xg x

x=

+ and the point P given by 0x = answer each of the following

questions.


(i) 1 (ii) 0.5 (iii) 0.1 (iv) 0.01 (v) 0.001 (vi) -1 (vii) -0.5 (viii) -0.1 (ix) -0.01 (x) -0.001


Calculus I


(b) Use the information from (a) to estimate the slope of the tangent line to ( )g x at 0x =


3. For the function ( ) ( )22 2h x x= − + and the point P given by 2x = − answer each of the



(i) -2.5 (ii) -2.1 (iii) -2.01 (iv) -2.001 (v) -2.0001 (vi) -1.5 (vii) -1.9 (viii) -1.99 (ix) -1.999 (x) -1.9999

(b) Use the information from (a) to estimate the slope of the tangent line to ( )h x at 2x = −


4. For the function ( ) 22 8xP x −= e and the point P given by 0.5x = answer each of the following

questions.


(i) 1 (ii) 0.51 (iii) 0.501 (iv) 0.5001 (v) 0.50001 (vi) 0 (vii) 0.49 (viii) 0.499 (ix) 0.4999 (x) 0.49999

(b) Use the information from (a) to estimate the slope of the tangent line to ( )h x at 0.5x =


5. The amount of grain in a bin is given by ( ) 11 44

tV tt

+=

+ answer each of the following

questions.

(a) Compute (accurate to at least 8 decimal places) the average rate of change of the amount of grain in the bin between 6t = and the following values of t.


(b) Use the information from (a) to estimate the instantaneous rate of change of the volume of air in the balloon at 6t = .


Calculus I


6. The population (in thousands) of insects is given by ( ) ( )12 cos 3 sin2tP t t π

ππ

= −

answer

each of the following questions.

(a) Compute (accurate to at least 8 decimal places) the average rate of change of the population of insects between 4t = and the following values of t. Make sure your calculator is set to radians for the computations.


(b) Use the information from (a) to estimate the instantaneous rate of change of the population of the insects at 4t = .

7. The amount of water in a holding tank is given by ( ) 4 28 7V t t t= − + answer each of the


(a) Compute (accurate to at least 8 decimal places) the average rate of change of the amount of grain in the bin between 0.25t = and the following values of t.

(i) 1 (ii) 0.5 (iii) 0.251 (iv) 0.2501 (v) 0.25001 (vi) 0 (vii) 0.1 (viii) 0.249 (ix) 0.2499 (x) 0.24999

(b) Use the information from (a) to estimate the instantaneous rate of change of the volume of water in the tank at 0.25t = .

8. The position of an object is given by ( ) 2 721

s t xx

= ++

answer each of the following

questions.

(a) Compute (accurate to at least 8 decimal places) the average velocity of the object between 5t = and the following values of t. (i) 5.5 (ii) 5.1 (iii) 5.01 (iv) 5.001 (v) 5.0001 (vi) 4.5 (vii) 4.9 (viii) 4.99 (ix) 4.999 (x) 4.9999

(b) Use the information from (a) to estimate the instantaneous velocity of the object at 5t = and determine if the object is moving to the right (i.e. the instantaneous velocity is positive), moving to the left (i.e. the instantaneous velocity is negative), or not moving (i.e. the instantaneous velocity is zero).

9. The position of an object is given by ( ) ( ) ( )2cos 4 7sins t t t= − . Note that a negative

position here simply means that the position is to the left of the “zero position” and is perfectly acceptable. Answer each of the following questions.


Calculus I


(a) Compute (accurate to at least 8 decimal places) the average velocity of the object between 0t = and the following values of t. Make sure your calculator is set to radians for the

computations. (i) 2.5 (ii) 2.1 (iii) 2.01 (iv) 2.001 (v) 2.0001 (vi) 1.5 (vii) 1.9 (viii) 1.99 (ix) 1.999 (x) 1.9999

(b) Use the information from (a) to estimate the instantaneous velocity of the object at 0t = and determine if the object is moving to the right (i.e. the instantaneous velocity is positive), moving to the left (i.e. the instantaneous velocity is negative), or not moving (i.e. the instantaneous velocity is zero).

10. The position of an object is given by ( ) 2 10 11s t t t= − + . Note that a negative position here

simply means that the position is to the left of the “zero position” and is perfectly acceptable. Answer each of the following questions.

(a) Determine the time(s) in which the position of the object is at 5s = −

(b) Estimate the instantaneous velocity of the object at each of the time(s) found in part (a) using the method discussed in this section.

The Limit

1. For the function ( )2

2

6 93

x xg xx x+ +

=+

answer each of the following questions.

(a) Evaluate the function the following values of x compute (accurate to at least 8 decimal places).

(i) -2.5 (ii) -2.9 (iii) -2.99 (iv) -2.999 (v) -2.9999 (vi) -3.5 (vii) -3.1 (viii) -3.01 (ix) -3.001 (x) -3.0001

(b) Use the information from (a) to estimate the value of 2

23

6 9lim3x

x xx x→−

+ ++

.

2. For the function ( )2

2

10 91

z zf zz− −

=−

answer each of the following questions.

(a) Evaluate the function the following values of t compute (accurate to at least 8 decimal places).

(i) 1.5 (ii) 1.1 (iii) 1.01 (iv) 1.001 (v) 1.0001


Calculus I


(vi) 0.5 (vii) 0.9 (viii) 0.99 (ix) 0.999 (x) 0.9999


21

10 9lim1z

z zz→

− −−

.

3. For the function ( ) 2 4 2th tt

− += answer each of the following questions.

(a) Evaluate the function the following values of θ compute (accurate to at least 8 decimal places). Make sure your calculator is set to radians for the computations.

(i) 0.5 (ii) 0.1 (iii) 0.01 (iv) 0.001 (v) 0.0001 (vi) -0.5 (vii) -0.1 (viii) -0.01 (ix) -0.001 (x) -0.0001


2 4 2limt

tt→

− +.

4. For the function ( ) ( )cos 4 12 8

gθ

θθ

− −=

− answer each of the following questions.

(a) Evaluate the function the following values of θ compute (accurate to at least 8 decimal places). Make sure your calculator is set to radians for the computations.


(b) Use the information from (a) to estimate the value of ( )

0

cos 4 1lim

2 8θ

θθ→

− −−

.

5. Below is the graph of ( )f x . For each of the given points determine the value of ( )f a and

( )limx a

f x→

. If any of the quantities do not exist clearly explain why.

(a) 2a = − (b) 1a = − (c) 2a = (d) 3a =


Calculus I



( )limx a

f x→


(a) 3a = − (b) 1a = − (c) 1a = (d) 3a =


( )limx a

f x→


(a) 4a = − (b) 2a = − (c) 1a = (d) 4a =


Calculus I


8. Explain in your own words what the following equation means. ( )

12lim 6x

f x→

=

9. Suppose we know that ( )

7lim 18x

f x→−

= . If possible, determine the value of ( )7f − . If it is

not possible to determine the value explain why not. 10. Is it possible to have ( )

1lim 23x

f x→

= − and ( )1 107f = ? Explain your answer.

One-Sided Limits

1. Below is the graph of ( )f x . For each of the given points determine the value of ( )f a ,

( )limx a

f x−→

, ( )limx a

f x+→

, and ( )limx a

f x→


(a) 5a = − (b) 2a = − (c) 1a = (d) 4a =


Calculus I



( )limx a

f x−→

, ( )limx a

f x+→

, and ( )limx a

f x→


(a) 1a = − (b) 1a = (c) 3a =


( )limx a

f x−→

, ( )limx a

f x+→

, and ( )limx a

f x→


(a) 3a = − (b) 1a = − (c) 1a = (d) 2a =

4. Sketch a graph of a function that satisfies each of the following conditions. ( ) ( ) ( )

1 1lim 2 lim 3 1 6x x

f x f x f− +→ →

= − = =

5. Sketch a graph of a function that satisfies each of the following conditions. ( ) ( ) ( )

3 3lim 1 lim 1 3 4

x xf x f x f

− +→− →−= = − =


Calculus I


6. Sketch a graph of a function that satisfies each of the following conditions.

( ) ( ) ( )

( ) ( )5 5

4

lim 1 lim 7 5 4

lim 6 4 does not existx x

x

f x f x f

f x f

− +→− →−

→

= − = − =

=

7. Explain in your own words what each of the following equations mean. ( ) ( )

8 8lim 3 lim 1x x

f x f x− +→ →

= = −

8. Suppose we know that ( )

7lim 18x

f x→−

= . If possible, determine the value of ( )7

limx

f x−→−

and

the value of ( )7

limx

f x+→−

. If it is not possible to determine one or both of these values explain

why not. 9. Suppose we know that ( )6 53f = − . If possible, determine the value of ( )

6limx

f x−→

and the

value of ( )6

limx

f x+→

. If it is not possible to determine one or both of these values explain why

not.

Limit Properties 1. Given ( )

0lim 5x

f x→

= , ( )0

lim 1x

g x→

= − and ( )0

lim 3x

h x→

= − use the limit properties given in this

section to compute each of the following limits. If it is not possible to compute any of the limits clearly explain why not. (a) ( )

0lim 11 7x

f x→

+ (b) ( ) ( )0

lim 6 4 10x

g x h x→

− −

(c) ( ) ( ) ( )0

lim 4 12 3x

g x f x h x→

− + (d) ( ) ( )( )0

lim 1 2x

g x f x→

+

2. Given ( )

12lim 2x

f x→

= , ( )12

lim 6x

g x→

= and ( )12

lim 9x

h x→

= use the limit properties given in this

section to compute each of the following limits. If it is not possible to compute any of the limits clearly explain why not.

(a) ( ) ( ) ( )( )12

1limx

g xh x f x

g x→

++

(b) ( )( ) ( )( )

12lim 3 1 2x

f x g x→

− +


Calculus I


(c) ( )

( ) ( )12

1lim

3 2x

f xg x h x→

+−

(d) ( ) ( )

( ) ( )12

2lim

7x

f x g xh x f x→

−+

3. Given ( )

1lim 0x

f x→−

= , ( )1

lim 9x

g x→−

= and ( )1

lim 7x

h x→−

= − use the limit properties given in this

section to compute each of the following limits. If it is not possible to compute any of the limits clearly explain why not.

(a) ( )( ) ( )( )2 3

1limx

g x h x→−

− (b) ( ) ( )1

lim 3 6x

f x h x→−

+ −

(c) ( ) ( ) ( )1

limx

f x g x h x→−

− (d) ( )( )

41

2lim

1 10x

g xh x→−

+−

For each of the following limits use the limit properties given in this section to compute the limit. At each step clearly indicate the property being used. If it is not possible to compute any of the limits clearly explain why not.

4. ( )2

4lim 3 9 2x

x x→

− +

5. ( )( )22

1lim 3w

w w→−

− +

6. ( )4 2

0lim 4 12 8t

t t t→

− + −

7. 2

2

10lim3 4z

zz→

+−

8. 27

8lim14 49x

xx x→ − +

9. 3

23

20 4lim8 1y

y yy y→−

− ++ −

10. 3

6lim 8 7

ww

→−+

11. ( )2

1lim 4 8 1t

t t→

− +


Calculus I


12. ( )4

8lim 3 8 9 2x

x x→

− + +

Computing Limits For problems 1 – 20 evaluate the limit, if it exists.

1. ( )3

9lim 1 4x

x→−

−

2. ( )4 3

1lim 6 7 12 25y

y y y→

− + +

3. 2

20

6lim3t

tt→

+−

4. 24

6lim2 3z

zz→ +

5. 22

2lim6 16w

ww w→−

+− −

6. 2

25

6 5lim2 15t

t tt t→−

+ ++ −

7. 2

23

5 16 3lim9x

x xx→

− +−

8. 2

21

10 9lim3 4 7z

z zz z→

− −+ −

9. 3

22

8lim8 12x

xx x→−

++ +

10. ( )

28

5 24lim

8t

t tt t→

− −−

11. ( )( )

2

4

16lim2 3 6w

ww w→−

−− + −


Calculus I


12. ( )3

0

2 8limh

hh→

+ −

13. ( )4

0

1 1limh

hh→

+ −

14. 25

5lim25t

tt→

−−

15. 2

2lim2x

xx→

−−

16. 6

6lim3 2 4z

zz→

−− −

17. 2

3 1 4lim2 4z

zz→−

− −+

18. 3

3lim1 5 11t

tt t→

−+ − −

19. 7

1 17lim

7x

xx→

−

−

20. 1

1 14 3lim

1y

y yy→−

++

+

21. Given the function

( ) 15 46 2 4

xf x

x x< −

= − ≥ −

Evaluate the following limits, if they exist. (a) ( )

7limx

f x→−

(b) ( )4

limx

f x→−



Calculus I


( )2 3 2

5 14 2t t t

g tt t

− <=

− ≥


3limt

g t→−

(b) ( )2

limt

g t→


( )22 68 6

w wh w

w w ≤

= − >


6limw

h w→

(b) ( )2

limw

h w→


( ) 2

5 24 33 4

1 2 4

x xg x x x

x x

+ < −= − ≤ < − ≥


3limx

g x→−

(b) ( )0

limx

g x→

(c) ( )4

limx

g x→

(d) ( )12

limx

g x→

For problems 25 – 30 evaluate the limit, if it exists.

25. ( )10

lim 10 3z

t→−

+ +

26. ( )4

lim 9 8 2x

x→

+ −

27. 0

limh

hh→

28. 2

2lim2t

tt→

−−

29. 5

2 10lim

5w

ww→−

++


Calculus I


30. 24

4lim

16x

xx→

−−

31. Given that ( )3 2 1x f x x+ ≤ ≤ − for all x determine the value of ( )

4limx

f x→−

.

32. Given that ( ) 172

xx f x −+ ≤ ≤ for all x determine the value of ( )

9limx

f x→

.

33. Use the Squeeze Theorem to determine the value of 4

0

3lim cosx

xx→

.

34. Use the Squeeze Theorem to determine the value of 0

1lim cosx

xx→

.

35. Use the Squeeze Theorem to determine the value of ( )2

1

1lim 1 cos1x

xx→

− − .

Infinite Limits For problems 1 – 8 evaluate the indicated limits, if they exist.

1. For ( )( )2

41

g xx−

=−

evaluate,

(a) ( )1

limx

g x−→

(b) ( )1

limx

g x+→

(c) ( )1

limx

g x→

2. For ( )( )3

174

h zz

=−

evaluate,

(a) ( )4

limz

h z−→

(b) ( )4

limz

h z+→

(c) ( )4

limz

h z→

3. For ( )( )

2

74

3tg t

t=

+ evaluate,

(a) ( )3

limt

g t−→−

(b) ( )3

limt

g t+→−

(c) ( )3

limt

g t→−


Calculus I


4. For ( ) 3

18

xf xx

+=

+ evaluate,

(a) ( )2

limx

f x−→−

(b) ( )2

limx

f x+→−

(c) ( )2

limx

f x→−

5. For ( )( )42

1

9

xf xx

−=

− evaluate,

(a) ( )3

limx

f x−→

(b) ( )3

limx

f x+→

(c) ( )3

limx

f x→

6. For ( ) ( )ln 8W t t= + evaluate,

(a) ( )8

limw

W t−→−

(b) ( )8

limw

W t+→−

(c) ( )8

limw

W t→−

7. For ( ) lnh z z= evaluate,

(a) ( )0

limz

h z−→

(b) ( )0

limz

h z+→

(c) ( )0

limz

h z→

8. For ( ) ( )cotR y y= evaluate,

(a) ( )limy

R yπ −→

(b) ( )limy

R yπ +→

(c) ( )limy

R yπ→

For problems 9 – 12 find all the vertical asymptotes of the given function.

9. ( ) 69

h xx

−=

−

10. ( )( )32

85 2xf x

x x+

=−

11. ( ) ( )( )5

7 12tg t

x x x=

+ −

12. ( )( ) ( )

2

5 62

1

1 15

zg zz z

+=

− +


Calculus I


Limits At Infinity, Part I 1. For ( ) 3 58 9 11f x x x x= + − evaluate each of the following limits.

(a) ( )limx

f x→−∞

(b) ( )limx

f x→∞

2. For ( ) 2 410 6 2h t t t t= + + − evaluate each of the following limits.

(a) ( )limt

h t→−∞

(b) ( )limt

h t→∞

3. For ( ) 3 47 8g z z z= + + evaluate each of the following limits.

(a) ( )limz

g z→−∞

(b) ( )limz

g z→∞

For problems 4 – 17 answer each of the following questions. (a) Evaluate ( )lim

xf x

→−∞

(b) Evaluate ( )limx

f x→∞

(c) Write down the equation(s) of any horizontal asymptotes for the function.

4. ( )3

3

10 67 9

x xf xx

−=

+

5. ( ) 2

123 8 23

xf xx x

+=

− +

6. ( )8

3 5 8

5 910 3xf x

x x x−

=+ −

7. ( )2

2

2 6 915 4

x xf xx x

− −=

+ −

8. ( )4

2

5 74x xf x

x+

=−

9. ( )3 2

3

4 3 2 110 5

x x xf xx x

− + −=

− +


Calculus I


10. ( )8

3

52 7 1

xf xx x

−=

− +

11. ( )3 21 4

9 10xf xx

+=

+

12. ( )2

25 75 2

xf xx

+=

+

13. ( )28 11

9xf xx

+=

− −

14. ( )4 2

2

9 2 35 2x xf x

x x+ +

=−

15. ( )3

6

68 4

xf xx

+=

+

16. ( )3 32 84 7

xf xx

−=

+

17. ( )44

15 2

xf xx

+=

+

Limits At Infinity, Part II

For problems 1 – 11 evaluate (a) ( )limx

f x→−∞

and (b) ( )limx

f x→∞

.

1. ( ) 4 8x xf x += e

2. ( ) 2 52 4 2x x xf x + += e

3. ( )3

23 xx xf x

−

+= e


Calculus I


4. ( )5 97 3

xxf x

−+= e

5. ( )6

45 2

8x

x xf x+

−= e

6. ( ) 3 1012 2x x xf x − −= + −e e e

7. ( ) 2 149 7x x xf x −= − −e e e

8. ( ) 8 75 220 3x x x xf x − −= − + −e e e e

9. ( )15

15

4

46

11 6

x x

x xf x−

−

+=

+e ee e

10. ( )3 10

7

9 42

x x x

x xf x−

−+ −

=−

e e ee e

11. ( )14 18

20 9

32

x x

x x xf x−

− −−

=− −e e

e e e

For problems 12 – 20 evaluate the given limit.

12. ( )2lim ln 5 12 6x

x x→∞

+ −

13. ( )5lim ln 5 7y

y→−∞

−

14. 3

3lim ln1 5x

xx→∞

+ +

15. 3

2

2 5lim ln4 3t

t tt→−∞

− +

16. 2

2

10 8lim ln1z

z zz→∞

+ −

17. ( )1 3lim tan 7 4x

x x−

→−∞+ −


Calculus I


18. ( )1 2 6lim tan 4w

w w−

→∞−

18. 3 2

1 4lim tan1 3t

t tt

−

→∞

+ +

19. 4

12 3

4lim tan3 5z

zz z

−

→−∞

+ +

Continuity

1. The graph of ( )f x is given below. Based on this graph determine where the function is

discontinuous.


discontinuous.


Calculus I



discontinuous.

For problems 4 – 13 using only Properties 1- 9 from the Limit Properties section, one-sided limit properties (if needed) and the definition of continuity determine if the given function is continuous or discontinuous at the indicated points.

4. ( ) 6 27 14

xf xx+

=−

(a) 3x = − , (b) 0x = , (c) 2x = ?

5. ( ) 2

225

yR yy

=−

(a) 5y = − , (b) 1y = − , (c) 3y = ?

6. ( ) 2

5 2012

zg zz z

−=

−

(a) 1z = − , (b) 0z = , (c) 4z = ?

7. ( ) 2

26 7

xW xx x

+=

+ −

(a) 7x = − , (b) 0x = , (c) 1x = ?

8. ( )22 1

4 6 1z z

h zz z

< −=

+ ≥ −

(a) 6z = − , (b) 1z = − ?


Calculus I


9. ( ) 2

00

xx xg x

x x + <

= ≥

e

(a) 0x = , (b) 4x = ?

10. ( ) 8 51 6 5

tZ t

t t<

= − ≥

(a) 0t = , (b) 5t = ?

11. ( )2

2 40 4

18 4

z zh z z

z z

+ < −= = − − > −

(a) 4z = − , (b) 2z = ?

12. ( )

2

2

1 23 2

2 7 2 70 7

7

x xx

f x x xx

x x

− < − == − < < = >

(a) 2x = , (b) 7x = ?

13. ( )

3 00 0

6 0 814 8

22 8

w ww

g w w ww

w w

< == + < < =

− >

(a) 0w = , (b) 8w = ? For problems 14 – 22 determine where the given function is discontinuous.

14. ( ) 2

11 22 13 7

xf xx x

−=

− −

15. ( ) 2

32 3 4

Q zz z

=+ −

16. ( )2

3 2

16

th tt t t

−=

+ +


Calculus I


17. ( ) ( )2

4 15cos 1z

zf z +=

+

18. ( ) ( )1

sin 1xh x

x x−

=−

19. ( ) 7

34 1xf x −=

−e

20. ( )2

1

1

2

w

w wR w −

+

=−e

e e

21. ( ) ( )cot 4g x x=

22. ( ) ( )secf t t=

For problems 23 – 27 use the Intermediate Value Theorem to show that the given equation has at least one solution in the indicated interval. Note that you are NOT asked to find the solution only show that at least one must exist in the indicated interval. 23. 3 41 7 0x x+ − = on [ ]4,8 24. 2 11 3z z+ = on [ ]15, 5− −

25. 2 15 0

8t t

t+ −

=−

on [ ]5,1−

26. ( ) ( )2 2ln 2 1 ln 4 0t t+ − − = on [ ]1, 2−

27. 3 210 5ww w −= + −e on [ ]0,4

For problems 28 – 33 assume that ( )f x is continuous everywhere unless otherwise indicated in

some way. From the given information is it possible to determine if there is a root of ( )f x in

the given interval? If it is possible to determine that there is a root in the given interval clearly explain how you know that a root must exist. If it is not possible to determine if there is a root in the interval


Calculus I


sketch a graph of two functions each of which meets the given information and one will have a root in the given interval and the other will not have a root in the given interval. 28. ( )5 12f − = and ( )0 3f = − on the interval [ ]5,0− .

29. ( )1 30f = and ( )9 6f = on the interval [ ]1,9 .

30. ( )20 100f = − and ( )40 100f = − on the interval [ ]20,40 .

31. ( )4 10f − = − , ( )5 17f = , ( )

1lim 2x

f x−→

= − , and ( )1

lim 4x

f x+→

= on the interval [ ]4,5− .

32. ( )8 2f − = , ( )1 23f = , ( )

4lim 35

xf x

−→−= , and ( )

4lim 1

xf x

+→−= on the interval [ ]8,1− .

33. ( )0 1f = − , ( )9 10f = , ( )

2lim 12x

f x−→

= − , and ( )2

lim 3x

f x+→

= − on the interval [ ]0,10 .

The Definition of the Limit Problems for this section have not yet been written.


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