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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.6 • Assess Your Understanding 151
6. True or False A function f has a limit L at infinity, if for
any given ε>0, there is a positive number M so that
whenever x > M, then | f (x) − L| > ε.
Skill Building
In Problems 7–12, for each limit, find the largest δ that “works” for
the given ε.
7. lim x→1
(2x) = 2, ε = 0.01 8. lim x→2
(−3x) =−6, ε = 0.01
9. lim(6x − 1) = 11 10. lim (2 − 3x) = 11
x→2 x→−3
ε = 1 ε = 1 2
3
(
11. lim − 1 ) (
x→2 2 x + 5 = 4 12. lim 3x + 1 )
= 3
x→ 5 2
6
ε = 0.01 ε = 0.3
13. For the function f (x) = 4x − 1, we have lim x→3
f (x) = 11.
For each ε>0, find a δ>0 so that
whenever 0 < |x − 3| 0, find a δ>0 so that
whenever 0 < |x + 2| 0, find a δ>0 so that
∣ whenever 0 < |x + 3| 0, find a δ>0 so that
∣ whenever 0 < |x − 2| 0, let δ ≤ . See TSM for
(c) d ≤ 0.00025
complete proof. 2
ε
(d) δ ≤ ε
4
20. Given any e > 0, let δ ≤ . See TSM for
complete proof. 3
}
.
ε
21. Given any e > 0, let δ ≤ . See
TSM for complete proof. 5
ε
22. Given any e > 0, let δ ≤ . See
TSM for complete proof. 2
⎧ ε ⎫
23. Given any e > 0, let δ ≤ min⎨1, ⎬ .
See TSM for complete proof. ⎩ 3 ⎭
⎧ ε ⎫
24. Given any e > 0, let δ ≤ min⎨1, ⎬ .
See TSM for complete proof. ⎩ 4 ⎭
⎧ ε ⎫
25. Given any e > 0, let δ ≤ min⎨1, 2 ⎬ .
See TSM for complete proof. ⎩ 7 ⎭
⎧ ε ⎫
26. Given any e > 0, let δ ≤ min⎨1, 15 ⎬ .
See TSM for complete proof. ⎩ 4 ⎭
27. Given any e > 0, let δ ≤ ε
3 . See TSM
for complete proof.
28. Given any e > 0, let d ≤ e. See TSM
for complete proof.
⎧ ε ⎫
29. Given any e > 0, let δ ≤ min⎨1, ⎬ .
See TSM for complete proof. ⎩ 3 ⎭
⎧ ε ⎫
30. Given any e > 0, let δ ≤ min⎨1, ⎬ .
See TSM for complete proof. ⎩ 19 ⎭
31. Given any e > 0, let d ≤ min{1, 6e}.
See TSM for complete proof.
⎧ ε ⎫
32. Given any e > 0, let δ ≤ min⎨1, 4 ⎬ .
See TSM for complete proof. ⎩ 5 ⎭
33. See TSM.
34. See TSM.
35. Given any e > 0, let
⎧ ε ⎫
δ ≤ min⎨1, 234 ⎬.
⎩ 7 ⎭
See TSM for complete proof.
⎧ ε ⎫
36. Given any e > 0, let δ ≤ min⎨1, ⎬ .
⎩ 5
See TSM for complete proof. ⎭
37. Given any e > 0, let d ≤ min{1, 26e}.
See TSM for complete proof.
38. See TSM.
ε
39. Given any e > 0, let δ ≤ . See
TSM for complete proof. 1 + | m|
40. Given any e > 0, let
⎧ ε ⎫
δ ≤ min⎨
1 ⎬
⎩ 3 , 3 . See TSM.
13 ⎭
41. x must be within 0.05 of 3.
42. x must be within approx. 0.087 of 3.
43. See TSM.
1
44. Given any e > 0, let N ≤− . See
TSM for complete proof. ε
1
45. Given any e > 0, let M ≥
2
ε
. See
TSM for complete proof.
46. N =− 10
47. See TSM.
Section 1.6 • Assess Your Understanding
151
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