Thomas Calculus 13th [Solutions]

86 Chapter 2 Limits and Continuity
2.5 CONTINUITY
1. No, discontinuous at x 2, not defined at x 2
2. No, discontinuous at x 3,
3. Continuous on [ 1, 3]
4. No, discontinuous at x 1,
1 lim g( x) g(3) 1.5
x 3
1.5 lim k( x) lim k( x) 0
x 1 x 1
5. (a) Yes (b) Yes, lim f ( x) 0
x 1
(c) Yes (d) Yes
6. (a) Yes, f (1) 1
(b) Yes, lim f ( x) 2
x 1
(c) No
(d) No
7. (a) No (b) No
8. [ 1, 0) (0, 1) (1, 2) (2, 3)
9. f (2) 0, since
10. f (1) should be changed to
lim f ( x) 2(2) 4 0 lim f ( x)
x 2 x 2
2 lim f ( x)
x 1
11. Nonremovable discontinuity at x 1 because lim f ( x ) fails to exist ( lim f ( x ) 1 and lim f ( x) 0).
x 1
x 1
x 1
Removable discontinuity at x 0 by assigning the number lim f ( x ) 0 to be the value of f (0) rather
x 0
than f (0) 1.
12. Nonremovable discontinuity at x 1 because lim f ( x ) fails to exist ( lim f ( x) 2 and lim f ( x) 1).
x 1
x 1 x 1
Removable discontinuity at x 2 by assigning the number lim f ( x ) 1to be the value of f (2) rather than
x 2
f (2) 2.
13. Discontinuous only when x 2 0 x 2 14. Discontinuous only when ( x 2) 0 x 2
2
15. Discontinuous only when x 4x 3 0 ( x 3)( x 1) 0 x 3 or x 1
2
16. Discontinuous only when x 3x 10 0 ( x 5)( x 2) 0 x 5 or x 2
17. Continuous everywhere. (| x 1| sin x defined for all x; limits exist and are equal to function values.)
18. Continuous everywhere. (| x | 1 0 for all x; limits exist and are equal to function values.)
2
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