Unformatted text preview: 334 Section 8.3
f (x) xa g(x) 44. (a) f o(g) as xa if lim 0 3. If p
1 0 1, then
1 Suppose f and g are both positive in some open interval containing a. Then f O(g) as xa if there is a
f (x) g(x) dx xp lim
c0
c dx xp lim
c0 x positive integer M for which close to a. M for x sufficiently lim
c0 p 1 1 p 1 c 1 c p 1 p 1 because ( p 1) 0. (b) From Section 5.5, we know that ES
ES h
4 b a 4 h M 180
(4) 4. If 0
1 0 p 1, then
1 where M is a bound for the absolute value of f [a, b]. Thus, (b
M a) 180 on 1 dx xp lim
c0
c dx xp
p 1 int (b M a) 180 lim
c0 x as h0, so ES O(h 4). Thus as h0, ES0.
b 12 a 2 h M lim
c0 1 p 1 c 1 c p 1 p 1 1 1 p (c) From Section 5.6, we know that ET
ET h2 Quick Review 8.3
3 where M is a bound for the absolute value of f on [a, b]. Thus h0, so ET 45. (a) lim
x 1.
0 dx x 3 x dx x
2 3 ln x 3
0 ln 6
1 ln 3
1 ln 2 2 ln 2
1 ln 2 2 (b
2 a) M 12 int (b a) M 12 1 as 2. 1 1 O(h ). Thus as h0, ET0. lim
x 1 1 4 1 ln x 2 2 dx
x 2 2 1
1 0 f(x) g(x) f (x) g(x) f (x) x g(x) lim 3. dx x
2 4 1 Thus f grows faster than g as x by definition. (b) lim
x f(x) g(x) lim
x f (x) g(x) f (x) lim x g(x) 1 x 2 tan 1 4 2 1 x tan 1 C 2 2 1 3 x 4 dx x 3 C L 4.
dx x4 Thus f grows at the same rate as g as x by definition. 46. (a) lim
x C 3 f ( x) g( x) lim
x f (x) g(x) 5. 9 x 2 0 for 3 x The domain is ( 3, 3). 6. x 1 0 for x 1 The domain is (1, ). 7. L 1
cos x x2 Thus f ( x) grows faster than g( x) by definition.
f ( x) (b) lim x g( x) f (x) lim g(x) x cos x
cos x x2 1, so cos x
1 x2 1. Thus f ( x) grows at the same rate as g(x) by definition. 8. x 2
1 x2 1
1 x 2 so
1 x x2 1 x2 x for x 1 s Section 8.3 Improper Integrals
(pp. 433444)
1 9. lim f (x) x g(x) 4e x x x 3e lim 5 7 4e x x x 3e lim lim
x 4 3 4 3 Exploration 1
1. Because
1 Investigating
0 dx xp 0. Thus f and g grow at the same rate as x . 1 has an infinite discontinuity at x xp
1 10. lim
x f (x) g(x) lim
x 2x x 2x x 2 1 3 1 3
1 x 3 x 2.
0 dx x lim
c0
c dx x lim ln x
c0 1 c lim ( ln c)
c0 lim
x lim
x 2 1 ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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