Practice Exercises

Chapter 2 Practice Exercises

Limits and Continuity

2. Repeat the instructions of Exercise 1 for 1. Graph the function

0, x…- 1

06 ƒxƒ61

1 >x,

1, x…- 1 ƒsxd = d

0, x= 1

ƒsxd = e 1, x= 0

3. Suppose that ƒ(t) and g (t) are defined for all t and that lim t:t 0

1, xÚ 1. ƒstd = - 7 and lim t:t 0 g std = 0 . Find the limit as t:t 0 of the

following functions.

Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at 2

b. sƒstdd continuities removable? Explain.

x=- 1, 0 , and 1. Are any of the dis-

a. 3ƒ(t)

c. ƒstd # g std

e. cos (g (t))

f. ƒ ƒstd ƒ

g. ƒstd + g std

h. 1 >ƒ(t)

Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley

Chapter 2 Practice Exercises

4. Suppose that ƒ(x) and g (x) are defined for all x and that

lim x 23. 2 lim - 4x + 8 lim 24. 1

3x 3 x:q x 2 - 7x + 1 x:0 of the following functions.

x:0 ƒsxd = 1 >2 and lim x:0 g sxd = 22 . Find the limits as

x : -q

26. lim x 4 +x a. 3 -g sxd b. g sxd # ƒsxd

25. lim x 2 - 7x

x : -q x+ 1 x:q 12x 3 + 128 c. ƒsxd + g sxd

x:q :x; for - 5 … x … 5.d

27. lim sin x ƒsxd sIf you have a grapher, try graphing the function cos x

d. 1 >ƒ(x)

e. x+ ƒsxd

f. x- 1

>xd - 1d near the origin to

sIf you have a grapher, try graphing

cos u -1

In Exercises 5 and 6, find the value that lim

g sxd must have if the

28. lim u

ƒsxd = xscos s1

x:0

given limit statements hold. “see” the limit at infinity.d 4 - g sxd

u:q

x 2 >3 +x -1 5. lim a x

x+ sin x x:q x >3 + cos x 7. On what intervals are the following functions continuous?

b=1

x : -4 axlim x :0

Continuous Extension

a. ƒsxd = x 1 >3

b. g sxd = x 3 >4

c. be extended to be continuous at

2 -2 2 >3 -1 >6 31. Can ƒsxd = xsx - 1d >ƒx -1ƒ

x= 1 or -1? Give reasons for your answers. (Graph the func- 8. On what intervals are the following functions continuous?

hsxd = x

d. ksxd = x

tion—you will find the graph interesting.) a. ƒsxd = tan x b. g sxd = csc x 32. Explain why the function ƒsxd = sin s1 >xd has no continuous ex- tension to x= 0.

hsxd = cos x ksxd = sin x c. x-p d. x

T In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace

Finding Limits

and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension,

In Exercises 9–16, find the limit or explain why it does not exist. can it be extended to be continuous from the right or left? If so, what x 2

9. lim - 4x + 4 do you think the extended function’s value should be? x 3 + 5x 2 - 14x

u 33. ƒsxd = x- 1 5 cos a. as x:0

x- 2x 4 , a=1 34. g sud = 4u - 2p , a=p >2

b. as x:2

x 10. lim x 5 +x 4 3 35. hstd = s 1+ ƒ t ƒd 1 >t , a=0 36. k sxd = , a=0 x + 2x +x

1-2 ƒxƒ a. as x:0

b. as x : -1

Roots

37. Let ƒsxd = x 3 -x-1. sx + hd 2 -x 2 sx + hd 2 -x 2 a. Show that ƒ has a zero between -1 and 2.

h 14. x :0

13. lim lim

h b. Solve the equation ƒsxd = 0 graphically with an error of

h :0

magnitude at most 10 -8 .

1 1 lim 2+x

- 2 s2 + xd 3 -8

c. It can be shown that the exact value of the solution in part (b) is

18 b + a 2 - 18 b In Exercises 17–20, find the limit of g (x) as x approaches the indi-

x :0

x :0

Evaluate this exact answer and compare it with the value you cated value.

found in part (b).

17. lim s4g sxdd 1 >3

18. lim =2

38. Let ƒsud = u - 2u + 2 .

x :0

a. Show that ƒ has a zero between -2 and 0. 3x 2 +1 =q

b. Solve the equation ƒsud = 0 graphically with an error of

x :1

g sxd

x : -2 2gsxd

magnitude at most 10 -4 .

c. It can be shown that the exact value of the solution in part (b) is

Limits at Infinity

19 1 >3 Find the limits in Exercises 21–30. aA 27 -1 b - aA 27 +1 b

Evaluate this exact answer and compare it with the value you

x:q

x : -q 5x 2 +7

found in part (b).

Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley

Chapter 2: Limits and Continuity