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CHAPTER 2 LIMITS AND CONTINUITY
2.1 RATES OF CHANGE AND TANGENTS TO CURVES
?f f(3) f(2) 28 9 ?f f(1) f(") 20
1. (a) ?x œ 3# œ 1 œ 19 (b) ?x œ 1 (1) œ # œ1
?g g(1) g(1) 1 1 ?g g(0)g(2) 04
2. (a) ?x œ 1 (1) œ 2 œ0 (b) ?x œ 0(2) œ # œ 2
?h h ˆ 341 ‰ h ˆ 14 ‰ 1 1 ?h h ˆ 1# ‰ h ˆ 16 ‰ 0 È3 3 È 3
3. (a) ?t œ 31 1 œ 1 œ 14 (b) ?t œ 11 œ 1 œ 1
4 4 # # 6 3
?g g(1) g(0) (2 1) (2 1) ?g g(1) g(1) (2 1) (2 ")
4. (a) ?t œ 10 œ 10 œ 12 (b) ?t œ 1 (1) œ #1 œ0
?R R(2) R(0) È 8 1 È 1 3"
5. ?) œ 20 œ # œ # œ1
?P P(2) P(1) (8 16 10)(" % &)
6. ?) œ 21 œ 1 œ22œ0
?y ˆa2 h b2 3 ‰ ˆ 2 2 3 ‰ 4 4h h2 3 1 4h h2
7. (a) ?x œ h œ h œ h œ 4 h. As h Ä 0, 4 h Ä 4 Ê at Pa2, 1b the slope is 4.
(b) y 1 œ 4ax 2b Ê y 1 œ 4x 8 Ê y œ 4x 7
?y ˆ 5 a1 h b 2 ‰ ˆ 5 1 2 ‰ 5 1 2h h2 4 2h h2
8. (a) ?x œ h œ h œ h œ 2 h. As h Ä 0, 2 h Ä 2 Ê at Pa1, 4b the
slope is 2.
(b) y 4 œ a2bax 1b Ê y 4 œ 2x 2 Ê y œ 2x 6
?y ˆa2 h b2 2 a 2 h b 3 ‰ ˆ 2 2 2 a 2 b 3 ‰ 4 4h h2 4 2h 3 a3b 2h h2
9. (a) ?x œ h œ h œ h œ 2 h. As h Ä 0, 2 h Ä 2 Ê at
Pa2, 3b the slope is 2.
(b) y a3b œ 2ax 2b Ê y 3 œ 2x 4 Ê y œ 2x 7.
?y ˆa1 h b2 4 a 1 h b ‰ ˆ 1 2 4 a 1 b ‰ 1 2h h2 4 4h a3b h2 2h
10. (a) ?x œ h œ h œ h œ h 2. As h Ä 0, h 2 Ä 2 Ê at
Pa1, 3b the slope is 2.
(b) y a3b œ a2bax 1b Ê y 3 œ 2x 2 Ê y œ 2x 1.
?y a2 h b 3 2 3 8 12h 4h2 h3 8 12h 4h2 h3
11. (a) ?x œ h œ h œ h œ 12 4h h2 . As h Ä 0, 12 4h h2 Ä 12, Ê at
Pa2, 8b the slope is 12.
(b) y 8 œ 12ax 2b Ê y 8 œ 12x 24 Ê y œ 12x 16.
?y 2 a1 h b3 ˆ 2 1 3 ‰ 2 1 3h 3h2 h3 1 3h 3h2 h3
12. (a) ?x œ h œ h œ h œ 3 3h h2 . As h Ä 0, 3 3h h2 Ä 3, Ê at
Pa1, 1b the slope is 3.
(b) y 1 œ a3bax 1b Ê y 1 œ 3x 3 Ê y œ 3x 4.
?y a1 hb3 12a1 hb ˆ13 12a"b‰ 1 3h 3h2 h3 12 12h a11b 9h 3h2 h3
13. (a) ?x œ h œ h œ h œ 9 3h h2 . As h Ä 0,
9 3h h Ä 9 Ê at Pa1, 11b the slope is 9.
2
(b) y a11b œ a9bax 1b Ê y 11 œ 9x 9 Ê y œ 9x 2.
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44 Chapter 2 Limits and Continuity
?y a2 h b 3 3 a 2 h b 2 4 ˆ 2 3 3 a 2 b 2 4 ‰ 8 12h 6h2 h3 12 12h 3h2 4 0 3h2 h3
14. (a) ?x œ h œ h œ h œ 3h h2 . As h Ä 0,
3h h Ä 0 Ê at Pa2, 0b the slope is 0.
2
(b) y 0 œ 0ax 2b Ê y œ 0.
?p
15. (a) Q Slope of PQ œ ?t
650 225
Q" (10ß 225) 20 10 œ 42.5 m/sec
650 375
Q# (14ß 375) 20 14 œ 45.83 m/sec
650 475
Q$ (16.5ß 475) 20 16.5 œ 50.00 m/sec
650 550
Q% (18ß 550) 20 18 œ 50.00 m/sec
(b) At t œ 20, the sportscar was traveling approximately 50 m/sec or 180 km/h.
?p
16. (a) Q Slope of PQ œ ?t
80 20
Q" (5ß 20) 10 5 œ 12 m/sec
80 39
Q# (7ß 39) 10 7 œ 13.7 m/sec
80 58
Q$ (8.5ß 58) 10 8.5 œ 14.7 m/sec
80 72
Q% (9.5ß 72) 10 9.5 œ 16 m/sec
(b) Approximately 16 m/sec
17. (a)
?p 174 62
(b) ?t œ 2004 2002 œ 112
# œ 56 thousand dollars per year
(c) The average rate of change from 2001 to 2002 is ??pt œ 62 27
20022 2001 œ 35 thousand dollars per year.
The average rate of change from 2002 to 2003 is ??pt œ 2003
111 62
2002 œ 49 thousand dollars per year.
So, the rate at which profits were changing in 2002 is approximatley "# a35 49b œ 42 thousand dollars per year.
18. (a) F(x) œ (x 2)/(x 2)
x 1.2 1.1 1.01 1.001 1.0001 1
F(x) 4.0 3.4 3.04 3.004 3.0004 3
?F 4.0 (3) ?F 3.4 (3)
?x œ 1.2 1 œ 5.0; ?x œ 1.1 1 œ 4.4;
?F 3.04 (3) ?F 3.004 (3)
?x œ 1.01 1 œ 4.04; ?x œ 1.001 1 œ 4.!!%;
?F 3.!!!% (3)
?x œ 1.0001 1 œ 4.!!!%;
(b) The rate of change of F(x) at x œ 1 is 4.
?g g(2) g(1) È È1.5 "
19. (a) ?x œ 21 œ #21" ¸ 0.414213 ?g
?x œ g(1.5) g(1)
1.5 1 œ 0.5 ¸ 0.449489
?g g(1 h) g(1) È1 h"
?x œ (1 h) 1 œ h
(b) g(x) œ Èx
1h 1.1 1.01 1.001 1.0001 1.00001 1.000001
È1 h 1.04880 1.004987 1.0004998 1.0000499 1.000005 1.0000005
ŠÈ1 h 1‹ /h 0.4880 0.4987 0.4998 0.499 0.5 0.5
(c) The rate of change of g(x) at x œ 1 is 0.5.
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Section 2.1 Rates of Change and Tangents to Curves 45
È1 h"
(d) The calculator gives lim œ "# .
hÄ! h
f(3) f(2)
"" "
20. (a) i) 32 œ 3 #
1 œ 6
1 œ "6
" "
T # #TT
2
f(T) f(2) 2T 2T
ii) T# œ T# œ #T
T# œ #T(T 2) œ #T(2 T) œ #"T , T Á 2
(b) T 2.1 2.01 2.001 2.0001 2.00001 2.000001
f(T) 0.476190 0.497512 0.499750 0.4999750 0.499997 0.499999
af(T) f(2)b/aT 2b 0.2381 0.2488 0.2500 0.2500 0.2500 0.2500
(c) The table indicates the rate of change is 0.25 at t œ 2.
(d) lim ˆ #T
" ‰
œ 4"
TÄ#
NOTE: Answers will vary in Exercises 21 and 22.
21. (a) Ò0, 1Ó: ˜ 15 0 ˜s 20 15 ˜s 30 20
˜t œ 1 0 œ 15 mphà Ò1, 2.5Ó: ˜t œ 2.5 1 œ 3 mphà Ò2.5, 3.5Ó: ˜t œ 3.5 2.5 œ 10 mph
s 10
(b) At Pˆ "# , 7.5‰: Since the portion of the graph from t œ 0 to t œ 1 is nearly linear, the instantaneous rate of change
" 15 7.5
will be almost the same as the average rate of change, thus the instantaneous speed at t œ # is 1 0.5 œ 15 mi/hr.
At Pa2, 20b: Since the portion of the graph from t œ 2 to t œ 2.5 is nearly linear, the instantaneous rate of change will
20
be nearly the same as the average rate of change, thus v œ 20
2.5 2 œ 0 mi/hr. For values of t less than 2, we have
?s
Q Slope of PQ œ ?t
15 20
Q" (1ß 15) 1 2 œ 5 mi/hr
19 20
Q# (1.5ß 19) 1.5 2 œ 2 mi/hr
19.9 20
Q$ (1.9ß 19.9) 1.9 2 œ 1 mi/hr
Thus, it appears that the instantaneous speed at t œ 2 is 0 mi/hr.
At Pa3, 22b:
Q Slope of PQ œ ? s
?t Q Slope of PQ œ ?s
?t
35 22 20 22
Q" (4ß 35) 43 œ 13 mi/hr Q" (2ß 20) 2 3 œ 2 mi/hr
30 22 20 22
Q# (3.5ß 30) 3.5 3 œ 16 mi/hr Q# (2.5ß 20) 2.5 3 œ 4 mi/hr
23 22 21.6 22
Q$ (3.1ß 23) 3.1 3 œ 10 mi/hr Q$ (2.9ß 21.6) 2.9 3 œ 4 mi/hr
Thus, it appears that the instantaneous speed at t œ 3 is about 7 mi/hr.
(c) It appears that the curve is increasing the fastest at t œ 3.5. Thus for Pa3.5, 30b
Q Slope of PQ œ ? s
?t Q Slope of PQ œ ?s
?t
35 30 22 30
Q" (4ß 35) 4 3.5 œ 10 mi/hr Q" (3ß 22) 3 3.5 œ 16 mi/hr
34 30 25 30
Q# (3.75ß 34) 3.75 3.5 œ 16 mi/hr Q# (3.25ß 25) 3.25 3.5 œ 20 mi/hr
32 30 28 30
Q$ (3.6ß 32) 3.6 3.5 œ 20 mi/hr Q$ (3.4ß 28) 3.4 3.5 œ 20 mi/hr
Thus, it appears that the instantaneous speed at t œ 3.5 is about 20 mi/hr.
˜A 10 15 ˜A 3.9 15 ˜A 0 1.4
22. (a) Ò0, 3Ó: ˜t œ 30 ¸ 1.67 day à
gal
Ò0, 5Ó: ˜t œ 50 ¸ 2.2 day à Ò7,
gal
10Ó: ˜t œ 10 7 ¸ 0.5 gal
day
(b) At Pa1, 14b:
?A ?A
Q Slope of PQ œ ?t Q Slope of PQ œ ?t
12.2 14 15 14
Q" (2ß 12.2) 2 1 œ 1.8 gal/day Q" (0ß 15) 0 1 œ 1 gal/day
13.2 14 14.6 14
Q# (1.5ß 13.2) 1.5 1 œ 1.6 gal/day Q# (0.5ß 14.6) 0.5 1 œ 1.2 gal/day
13.85 14 14.86 14
Q$ (1.1ß 13.85) 1.1 1 œ 1.5 gal/day Q$ (0.9ß 14.86) 0.9 1 œ 1.4 gal/day
Thus, it appears that the instantaneous rate of consumption at t œ 1 is about 1.45 gal/day.
At Pa4, 6b:
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46 Chapter 2 Limits and Continuity
?A ?A
Q Slope of PQ œ ?t Q Slope of PQ œ ?t
3.9 6 10 6
Q" (5ß 3.9) 54 œ 2.1 gal/day Q" (3ß 10) 3 4 œ 4 gal/day
4.8 6 7.8 6
Q# (4.5ß 4.8) 4.5 4 œ 2.4 gal/day Q# (3.5ß 7.8) 3.5 4 œ 3.6 gal/day
5.7 6 6.3 6
Q$ (4.1ß 5.7) 4.1 4 œ 3 gal/day Q$ (3.9ß 6.3) 3.9 4 œ 3 gal/day
Thus, it appears that the instantaneous rate of consumption at t œ 1 is 3 gal/day.
At Pa8, 1b:
Q Slope of PQ œ ??At Q Slope of PQ œ ?A
?t
0.5 1 1.4 1
Q" (9ß 0.5) 9 8 œ 0.5 gal/day Q" (7ß 1.4) 7 8 œ 0.6 gal/day
0.7 1 1.3 1
Q# (8.5ß 0.7) 8.5 8 œ 0.6 gal/day Q# (7.5ß 1.3) 7.5 8 œ 0.6 gal/day
0.95 1 1.04 1
Q$ (8.1ß 0.95) 8.1 8 œ 0.5 gal/day Q$ (7.9ß 1.04) 7.9 8 œ 0.6 gal/day
Thus, it appears that the instantaneous rate of consumption at t œ 1 is 0.55 gal/day.
(c) It appears that the curve (the consumption) is decreasing the fastest at t œ 3.5. Thus for Pa3.5, 7.8b
Q Slope of PQ œ ??At Q Slope of PQ œ ? s
?t
4.8 7.8 11.2 7.8
Q" (4.5ß 4.8) 4.5 3.5 œ 3 gal/day Q" (2.5ß 11.2) 2.5 3.5 œ 3.4 gal/day
6 7.8 10 7.8
Q# (4ß 6) 4 3.5 œ 3.6 gal/day Q# (3ß 10) 3 3.5 œ 4.4 gal/day
7.4 7.8 8.2 7.8
Q$ (3.6ß 7.4) 3.6 3.5 œ 4 gal/day Q$ (3.4ß 8.2) 3.4 3.5 œ 4 gal/day
Thus, it appears that the rate of consumption at t œ 3.5 is about 4 gal/day.
2.2 LIMIT OF A FUNCTION AND LIMIT LAWS
1. (a) Does not exist. As x approaches 1 from the right, g(x) approaches 0. As x approaches 1 from the left, g(x)
approaches 1. There is no single number L that all the values g(x) get arbitrarily close to as x Ä 1.
(b) 1 (c) 0 (d) 0.5
2. (a) 0
(b) 1
(c) Does not exist. As t approaches 0 from the left, f(t) approaches 1. As t approaches 0 from the right, f(t)
approaches 1. There is no single number L that f(t) gets arbitrarily close to as t Ä 0.
(d) 1
3. (a) True (b) True (c) False
(d) False (e) False (f) True
(g) True
4. (a) False (b) False (c) True
(d) True (e) True
5. lim x
does not exist because x
œ x
œ 1 if x 0 and x
œ x
œ 1 if x 0. As x approaches 0 from the left,
x Ä 0 kx k kx k x kxk x
x
kx k approaches 1. As x approaches 0 from the right, x
kx k approaches 1. There is no single number L that all the
function values get arbitrarily close to as x Ä 0.
"
6. As x approaches 1 from the left, the values of x 1 become increasingly large and negative. As x approaches 1
from the right, the values become increasingly large and positive. There is no one number L that all the function
values get arbitrarily close to as x Ä 1, so lim x" 1 does not exist.
xÄ1
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CHAPTER 2 LIMITS AND CONTINUITY
2.1 RATES OF CHANGE AND TANGENTS TO CURVES
?f f(3) f(2) 28 9 ?f f(1) f(") 20
1. (a) ?x œ 3# œ 1 œ 19 (b) ?x œ 1 (1) œ # œ1
?g g(1) g(1) 1 1 ?g g(0)g(2) 04
2. (a) ?x œ 1 (1) œ 2 œ0 (b) ?x œ 0(2) œ # œ 2
?h h ˆ 341 ‰ h ˆ 14 ‰ 1 1 ?h h ˆ 1# ‰ h ˆ 16 ‰ 0 È3 3 È 3
3. (a) ?t œ 31 1 œ 1 œ 14 (b) ?t œ 11 œ 1 œ 1
4 4 # # 6 3
?g g(1) g(0) (2 1) (2 1) ?g g(1) g(1) (2 1) (2 ")
4. (a) ?t œ 10 œ 10 œ 12 (b) ?t œ 1 (1) œ #1 œ0
?R R(2) R(0) È 8 1 È 1 3"
5. ?) œ 20 œ # œ # œ1
?P P(2) P(1) (8 16 10)(" % &)
6. ?) œ 21 œ 1 œ22œ0
?y ˆa2 h b2 3 ‰ ˆ 2 2 3 ‰ 4 4h h2 3 1 4h h2
7. (a) ?x œ h œ h œ h œ 4 h. As h Ä 0, 4 h Ä 4 Ê at Pa2, 1b the slope is 4.
(b) y 1 œ 4ax 2b Ê y 1 œ 4x 8 Ê y œ 4x 7
?y ˆ 5 a1 h b 2 ‰ ˆ 5 1 2 ‰ 5 1 2h h2 4 2h h2
8. (a) ?x œ h œ h œ h œ 2 h. As h Ä 0, 2 h Ä 2 Ê at Pa1, 4b the
slope is 2.
(b) y 4 œ a2bax 1b Ê y 4 œ 2x 2 Ê y œ 2x 6
?y ˆa2 h b2 2 a 2 h b 3 ‰ ˆ 2 2 2 a 2 b 3 ‰ 4 4h h2 4 2h 3 a3b 2h h2
9. (a) ?x œ h œ h œ h œ 2 h. As h Ä 0, 2 h Ä 2 Ê at
Pa2, 3b the slope is 2.
(b) y a3b œ 2ax 2b Ê y 3 œ 2x 4 Ê y œ 2x 7.
?y ˆa1 h b2 4 a 1 h b ‰ ˆ 1 2 4 a 1 b ‰ 1 2h h2 4 4h a3b h2 2h
10. (a) ?x œ h œ h œ h œ h 2. As h Ä 0, h 2 Ä 2 Ê at
Pa1, 3b the slope is 2.
(b) y a3b œ a2bax 1b Ê y 3 œ 2x 2 Ê y œ 2x 1.
?y a2 h b 3 2 3 8 12h 4h2 h3 8 12h 4h2 h3
11. (a) ?x œ h œ h œ h œ 12 4h h2 . As h Ä 0, 12 4h h2 Ä 12, Ê at
Pa2, 8b the slope is 12.
(b) y 8 œ 12ax 2b Ê y 8 œ 12x 24 Ê y œ 12x 16.
?y 2 a1 h b3 ˆ 2 1 3 ‰ 2 1 3h 3h2 h3 1 3h 3h2 h3
12. (a) ?x œ h œ h œ h œ 3 3h h2 . As h Ä 0, 3 3h h2 Ä 3, Ê at
Pa1, 1b the slope is 3.
(b) y 1 œ a3bax 1b Ê y 1 œ 3x 3 Ê y œ 3x 4.
?y a1 hb3 12a1 hb ˆ13 12a"b‰ 1 3h 3h2 h3 12 12h a11b 9h 3h2 h3
13. (a) ?x œ h œ h œ h œ 9 3h h2 . As h Ä 0,
9 3h h Ä 9 Ê at Pa1, 11b the slope is 9.
2
(b) y a11b œ a9bax 1b Ê y 11 œ 9x 9 Ê y œ 9x 2.
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44 Chapter 2 Limits and Continuity
?y a2 h b 3 3 a 2 h b 2 4 ˆ 2 3 3 a 2 b 2 4 ‰ 8 12h 6h2 h3 12 12h 3h2 4 0 3h2 h3
14. (a) ?x œ h œ h œ h œ 3h h2 . As h Ä 0,
3h h Ä 0 Ê at Pa2, 0b the slope is 0.
2
(b) y 0 œ 0ax 2b Ê y œ 0.
?p
15. (a) Q Slope of PQ œ ?t
650 225
Q" (10ß 225) 20 10 œ 42.5 m/sec
650 375
Q# (14ß 375) 20 14 œ 45.83 m/sec
650 475
Q$ (16.5ß 475) 20 16.5 œ 50.00 m/sec
650 550
Q% (18ß 550) 20 18 œ 50.00 m/sec
(b) At t œ 20, the sportscar was traveling approximately 50 m/sec or 180 km/h.
?p
16. (a) Q Slope of PQ œ ?t
80 20
Q" (5ß 20) 10 5 œ 12 m/sec
80 39
Q# (7ß 39) 10 7 œ 13.7 m/sec
80 58
Q$ (8.5ß 58) 10 8.5 œ 14.7 m/sec
80 72
Q% (9.5ß 72) 10 9.5 œ 16 m/sec
(b) Approximately 16 m/sec
17. (a)
?p 174 62
(b) ?t œ 2004 2002 œ 112
# œ 56 thousand dollars per year
(c) The average rate of change from 2001 to 2002 is ??pt œ 62 27
20022 2001 œ 35 thousand dollars per year.
The average rate of change from 2002 to 2003 is ??pt œ 2003
111 62
2002 œ 49 thousand dollars per year.
So, the rate at which profits were changing in 2002 is approximatley "# a35 49b œ 42 thousand dollars per year.
18. (a) F(x) œ (x 2)/(x 2)
x 1.2 1.1 1.01 1.001 1.0001 1
F(x) 4.0 3.4 3.04 3.004 3.0004 3
?F 4.0 (3) ?F 3.4 (3)
?x œ 1.2 1 œ 5.0; ?x œ 1.1 1 œ 4.4;
?F 3.04 (3) ?F 3.004 (3)
?x œ 1.01 1 œ 4.04; ?x œ 1.001 1 œ 4.!!%;
?F 3.!!!% (3)
?x œ 1.0001 1 œ 4.!!!%;
(b) The rate of change of F(x) at x œ 1 is 4.
?g g(2) g(1) È È1.5 "
19. (a) ?x œ 21 œ #21" ¸ 0.414213 ?g
?x œ g(1.5) g(1)
1.5 1 œ 0.5 ¸ 0.449489
?g g(1 h) g(1) È1 h"
?x œ (1 h) 1 œ h
(b) g(x) œ Èx
1h 1.1 1.01 1.001 1.0001 1.00001 1.000001
È1 h 1.04880 1.004987 1.0004998 1.0000499 1.000005 1.0000005
ŠÈ1 h 1‹ /h 0.4880 0.4987 0.4998 0.499 0.5 0.5
(c) The rate of change of g(x) at x œ 1 is 0.5.
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Section 2.1 Rates of Change and Tangents to Curves 45
È1 h"
(d) The calculator gives lim œ "# .
hÄ! h
f(3) f(2)
"" "
20. (a) i) 32 œ 3 #
1 œ 6
1 œ "6
" "
T # #TT
2
f(T) f(2) 2T 2T
ii) T# œ T# œ #T
T# œ #T(T 2) œ #T(2 T) œ #"T , T Á 2
(b) T 2.1 2.01 2.001 2.0001 2.00001 2.000001
f(T) 0.476190 0.497512 0.499750 0.4999750 0.499997 0.499999
af(T) f(2)b/aT 2b 0.2381 0.2488 0.2500 0.2500 0.2500 0.2500
(c) The table indicates the rate of change is 0.25 at t œ 2.
(d) lim ˆ #T
" ‰
œ 4"
TÄ#
NOTE: Answers will vary in Exercises 21 and 22.
21. (a) Ò0, 1Ó: ˜ 15 0 ˜s 20 15 ˜s 30 20
˜t œ 1 0 œ 15 mphà Ò1, 2.5Ó: ˜t œ 2.5 1 œ 3 mphà Ò2.5, 3.5Ó: ˜t œ 3.5 2.5 œ 10 mph
s 10
(b) At Pˆ "# , 7.5‰: Since the portion of the graph from t œ 0 to t œ 1 is nearly linear, the instantaneous rate of change
" 15 7.5
will be almost the same as the average rate of change, thus the instantaneous speed at t œ # is 1 0.5 œ 15 mi/hr.
At Pa2, 20b: Since the portion of the graph from t œ 2 to t œ 2.5 is nearly linear, the instantaneous rate of change will
20
be nearly the same as the average rate of change, thus v œ 20
2.5 2 œ 0 mi/hr. For values of t less than 2, we have
?s
Q Slope of PQ œ ?t
15 20
Q" (1ß 15) 1 2 œ 5 mi/hr
19 20
Q# (1.5ß 19) 1.5 2 œ 2 mi/hr
19.9 20
Q$ (1.9ß 19.9) 1.9 2 œ 1 mi/hr
Thus, it appears that the instantaneous speed at t œ 2 is 0 mi/hr.
At Pa3, 22b:
Q Slope of PQ œ ? s
?t Q Slope of PQ œ ?s
?t
35 22 20 22
Q" (4ß 35) 43 œ 13 mi/hr Q" (2ß 20) 2 3 œ 2 mi/hr
30 22 20 22
Q# (3.5ß 30) 3.5 3 œ 16 mi/hr Q# (2.5ß 20) 2.5 3 œ 4 mi/hr
23 22 21.6 22
Q$ (3.1ß 23) 3.1 3 œ 10 mi/hr Q$ (2.9ß 21.6) 2.9 3 œ 4 mi/hr
Thus, it appears that the instantaneous speed at t œ 3 is about 7 mi/hr.
(c) It appears that the curve is increasing the fastest at t œ 3.5. Thus for Pa3.5, 30b
Q Slope of PQ œ ? s
?t Q Slope of PQ œ ?s
?t
35 30 22 30
Q" (4ß 35) 4 3.5 œ 10 mi/hr Q" (3ß 22) 3 3.5 œ 16 mi/hr
34 30 25 30
Q# (3.75ß 34) 3.75 3.5 œ 16 mi/hr Q# (3.25ß 25) 3.25 3.5 œ 20 mi/hr
32 30 28 30
Q$ (3.6ß 32) 3.6 3.5 œ 20 mi/hr Q$ (3.4ß 28) 3.4 3.5 œ 20 mi/hr
Thus, it appears that the instantaneous speed at t œ 3.5 is about 20 mi/hr.
˜A 10 15 ˜A 3.9 15 ˜A 0 1.4
22. (a) Ò0, 3Ó: ˜t œ 30 ¸ 1.67 day à
gal
Ò0, 5Ó: ˜t œ 50 ¸ 2.2 day à Ò7,
gal
10Ó: ˜t œ 10 7 ¸ 0.5 gal
day
(b) At Pa1, 14b:
?A ?A
Q Slope of PQ œ ?t Q Slope of PQ œ ?t
12.2 14 15 14
Q" (2ß 12.2) 2 1 œ 1.8 gal/day Q" (0ß 15) 0 1 œ 1 gal/day
13.2 14 14.6 14
Q# (1.5ß 13.2) 1.5 1 œ 1.6 gal/day Q# (0.5ß 14.6) 0.5 1 œ 1.2 gal/day
13.85 14 14.86 14
Q$ (1.1ß 13.85) 1.1 1 œ 1.5 gal/day Q$ (0.9ß 14.86) 0.9 1 œ 1.4 gal/day
Thus, it appears that the instantaneous rate of consumption at t œ 1 is about 1.45 gal/day.
At Pa4, 6b:
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46 Chapter 2 Limits and Continuity
?A ?A
Q Slope of PQ œ ?t Q Slope of PQ œ ?t
3.9 6 10 6
Q" (5ß 3.9) 54 œ 2.1 gal/day Q" (3ß 10) 3 4 œ 4 gal/day
4.8 6 7.8 6
Q# (4.5ß 4.8) 4.5 4 œ 2.4 gal/day Q# (3.5ß 7.8) 3.5 4 œ 3.6 gal/day
5.7 6 6.3 6
Q$ (4.1ß 5.7) 4.1 4 œ 3 gal/day Q$ (3.9ß 6.3) 3.9 4 œ 3 gal/day
Thus, it appears that the instantaneous rate of consumption at t œ 1 is 3 gal/day.
At Pa8, 1b:
Q Slope of PQ œ ??At Q Slope of PQ œ ?A
?t
0.5 1 1.4 1
Q" (9ß 0.5) 9 8 œ 0.5 gal/day Q" (7ß 1.4) 7 8 œ 0.6 gal/day
0.7 1 1.3 1
Q# (8.5ß 0.7) 8.5 8 œ 0.6 gal/day Q# (7.5ß 1.3) 7.5 8 œ 0.6 gal/day
0.95 1 1.04 1
Q$ (8.1ß 0.95) 8.1 8 œ 0.5 gal/day Q$ (7.9ß 1.04) 7.9 8 œ 0.6 gal/day
Thus, it appears that the instantaneous rate of consumption at t œ 1 is 0.55 gal/day.
(c) It appears that the curve (the consumption) is decreasing the fastest at t œ 3.5. Thus for Pa3.5, 7.8b
Q Slope of PQ œ ??At Q Slope of PQ œ ? s
?t
4.8 7.8 11.2 7.8
Q" (4.5ß 4.8) 4.5 3.5 œ 3 gal/day Q" (2.5ß 11.2) 2.5 3.5 œ 3.4 gal/day
6 7.8 10 7.8
Q# (4ß 6) 4 3.5 œ 3.6 gal/day Q# (3ß 10) 3 3.5 œ 4.4 gal/day
7.4 7.8 8.2 7.8
Q$ (3.6ß 7.4) 3.6 3.5 œ 4 gal/day Q$ (3.4ß 8.2) 3.4 3.5 œ 4 gal/day
Thus, it appears that the rate of consumption at t œ 3.5 is about 4 gal/day.
2.2 LIMIT OF A FUNCTION AND LIMIT LAWS
1. (a) Does not exist. As x approaches 1 from the right, g(x) approaches 0. As x approaches 1 from the left, g(x)
approaches 1. There is no single number L that all the values g(x) get arbitrarily close to as x Ä 1.
(b) 1 (c) 0 (d) 0.5
2. (a) 0
(b) 1
(c) Does not exist. As t approaches 0 from the left, f(t) approaches 1. As t approaches 0 from the right, f(t)
approaches 1. There is no single number L that f(t) gets arbitrarily close to as t Ä 0.
(d) 1
3. (a) True (b) True (c) False
(d) False (e) False (f) True
(g) True
4. (a) False (b) False (c) True
(d) True (e) True
5. lim x
does not exist because x
œ x
œ 1 if x 0 and x
œ x
œ 1 if x 0. As x approaches 0 from the left,
x Ä 0 kx k kx k x kxk x
x
kx k approaches 1. As x approaches 0 from the right, x
kx k approaches 1. There is no single number L that all the
function values get arbitrarily close to as x Ä 0.
"
6. As x approaches 1 from the left, the values of x 1 become increasingly large and negative. As x approaches 1
from the right, the values become increasingly large and positive. There is no one number L that all the function
values get arbitrarily close to as x Ä 1, so lim x" 1 does not exist.
xÄ1
Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.
