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1 2005/04/4 by. : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2. : : : : : : : : : : : : : : : : : : : : : : 3 3. : : : : : : : : : : : : : : : : : : : : : : : : : 4 4. : : : : : : : : : : : : : : : 5 5. A : : : : : : : : : : : : : : : : 6 6. B : : : : : : : : : : : : : : : : 7 7. : : : : : : : : : : : : : : : : : : : : : : 8 8. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 0 0. : : : : : : : : : : : : : : : : : : : : :. : : : : : : : : : : : : : : : : : : : : : 2. (A): (B): (C):

2 . a ;a 2 ;:::;a ;::: fa g; fa ;a 2 ;:::;a ;:::g; fa g =;2;::: ; fa g 2N a a N atural umbers. (A) () ; 2; 3; 4; 5; 6; 7; 8;::: (2) ; 4; 7; 0; 3; 6; 9;::: (3) 2; 4; 8; 6; 32; 64; 28;::: (4) ; =2; =3; =4; =5;::: (5) =2; =4; =8; =6; =32;::: (6) 0:9; 0:99; 0:999; 0:9999; 0:99999;::: (7) 4=3; 7=6; 0=9; 3=2;::: (8) ; =3; =9; =27; =8;::: (9) =5; =25; =25; =625;::: 2. (A) () (2)

3 2. fa g a ff fa g ff lim a = ff; a! ff (!) fa g (I) lim a = ff, lim b = fi () lim (ka + hb )=kff + hfi (2) lim (a b )=fffi (3) lim = a ff ( ff 6= 0) (4) a» b ff» fi (5) a» c» b, ff = fi lim c = ff (II) lim =0. ff fa g a + = a ff ; a > lim a a + = a ff b =loga b + = ffb fb g ff b = ff b. log a = ff log a a = a ff. 4 ffl jffj < ff! 0. a! a 0 =. ffl ff = ff =. a = a. ffl ff» ff a ffl ff> ff a X = e log X 3. (A) fa g p () a =+ 3 (2) a =+( ) (3) a = 0 (4) a = (7) a = (5) a = p + p (6) a = ( ) 2 (8) a = (9) a = p

4 3. x2.(i) fa g, fb g () (5) 3 lim a =, lim b = lim (a b ) (I)() k =, h = ffl a =2, b = lim (a b )=. ffl a = +,b = lim (a b )=. ffl a = p +, b = p lim (a b )=0. 0 ; ; 0 0 ; 0 ; 0 ; :. 0! a! 0, b! a b (I)(2) 0 3 ffl a =, b = lim (a b )= lim =. ffl a =, b = 2 lim (a b )= lim =. ffl a =, b 2 = lim (a b )= lim =0. 4. (A)! a!, b! a b a lim =0; b a lim = p; b a lim = ; b a lim = b fa g, fb g (p ) 4

5 4. fa g () (2) lim a () M a» a 2»»a»»M. (2) N a a 2 a N. e a = 2 + a = + = + ( ) + 2! = ++ 2! + +! a + = ++ 2! a <a +. a = + + +! 2 + 3! + + ( +)! ( )( 2) + 3! ! (Λ) < 2+ 2! + 3! + +! < < 3 2 (Λ) k! > 2 k (k 3) fa g () lim e (L. Euler, ) 5. (B) 5

6 5. A. a =,a + = 2a +3 a +2 fa g ()(2) lim a () a» a + ( =; 2; 3;:::) (2) a < 2( =; 2; 3;:::) ()(2) y y = x y = 2x +3 x +2 O x () a > 0 a =, a 2 = 5 3. a» a 2. = k a k» a k+ a k+ a k+2 = 2a k +3 a k +2 2a k+ +3 a k+ +2 = a k a k+ (a k +)(a k+ +)» 0: a k+» a k+2. a» a + (2) a =< 2 a > 0 a + =2 a < +2 2 ()(2) () fa g ff = lim a ff = 2ff +3 ff +2. ff> ff p = (A) (2) a < (A) fa g () a > 0; a + = pa + q (jpj < ; =; 2; 3;:::) (2) a = p; a 2 = q; a + = a + a ( =2; 3; 4;:::) 2 (3) a > 0; a p + = a ( =; 2; 3;:::) () 2 (2) 2 3 (3) 2 ()(2)(3) 6

7 6. B. a =,a p + = a +3 fa g ()(2) lim a () a» a + ( =; 2; 3;:::) (2) a < 5 2 ( =; 2; 3;:::) ()(2) y p y = x +3 y = x 3 O x () a =,a 2 =2 a» a 2. = k a k» a k+ a k+ a k+2 = p a k + p a k+ +» p a k+ + p a k+ +=0 a k+» a k+2. a» a + (2) a = < 5=2. = k a k < 5=2 a p p k+ = a k +3< 5=2 +3< 5=2. a < 5=2 ()(2) () fa g ff = lim a ff 2 = ff +3. <ff» 5=2 ff = +p (A) (2) a < (A) fa g a > 0; a + = p a +3 ( =; 2; 3;:::) 0. (B) fa g () a =; a + = a + p=a 2 (p >0; =; 2; 3;:::) (2) 0 <a < 2; a + = 3 4 a a ( =; 2; 3;:::) () p 7

8 7. 2 fa g, fb g () a» a 2»»a»»b»»b 2» b (2) lim (b a )=0 fa g fb g lim a = lim b.. I =[a;b] I+ ρ I 2 jij. a =, a + = ( = ; 2; 3;:::) fa g +a! a > a () a 2 a 2 ( =; 2; 3;:::) = I (2) a 2» a» a (3) a + a! 0(!) ()(2)(3) y y = x y = +x O x () a 4 a 2 > 0 a 2 a 2 2 > 0 a 2+2 a 2 > 0 a 3 a > 0 a 2 a 2 3 > 0 a 2+ a 2 > 0 (2) a > 0 a + < =a a + = = a 2. +a +a (3) ja + a j» ja a j ( + a )( + a» ja a j»» ja 2 a j! 0(!) ) ( + a 2 ) 2 ( + a 2 ) 2( ) 8

9 8.. (B) fa g () a > 0; a + =+ 2 2+a ( =; 2; 3;:::) (2) a =; a + =+ a ( =; 2; 3;:::) y y = x y =+ 2 2+x O 0:5 x y y = x y =+ x O x 2. (C) a = log, b =+ + log ( +) fa g fb g lim (a b )=0. 2 fa g, fb g fl =0: fl e + ß 9

10 lim (a b )=0 fa g fb g 3. (A) + < log + < ( =; 2;:::) (9.) a a + > 0, b b > 0 2. (9.) 4. (A) + <e< + + ( =; 2;:::) (9.2) (9.) a = +, b = + (9.) a <e<b + a <e x4. b = + a b >a, lim b = e b =4>e fb g (9.2) 5. (A) + > ( =; 2;:::) (9.3) + b <b (9.3) 2, a ;a 2 ;:::;a > 0 a + a a p a a 2 a a = a 2 = = a 6. (A) a =,a 2 = = a + = (9.3) x > + a =,a 2 = = a = x4. a >a 0

11 0. x9. 0 < a» a 2»»a A = a + a a =2 >2 OK a 0 = a + a A arithmetic mea a 0 + a a q a 0 a 2 a (0.) 7. (A) (0.) ( ) ( ) a 0 a a A (0.2) A a a (0.2) 8. (A) (0.2) A (( ) ( )) 0 a = = a 9. (A) a <a (0.2) a <a 20. (C) () a > 0, b > 0 x9. a a + b p + = ; b + = a b ( =; 2; 3;:::) 2 fa g, fb g a, b a, b ( 2) fa g fb g lim b ) = 0 Gauss 2 (2) a>0, b>0, a = 2 (a + b), b = p a b a = a + b p ; b = a b ( =2; 3;:::) 2 fa g, fb g

12 . : =6; 2; 24; 48; 96 a, b A ;B a b ß A B ß : Archimedes(287? 22 B.C.) a 96, b 96 (263) 3072 ß ο 3:459 ( ) ß ο 3: =3: ß (B) () a 6 =3,b 6 =2 p 3 2a (2) a 2 = p +a = + p a = ( =3; 4; 5 :::) (3) b 2 = 2b ( =3; 4; 5 :::) + p +(b =) 2 (4) a 2 >a, b 2 <b (5) A ;B a ;b ; (6) ()(2)(4) =6; 2 2

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