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