infi-07b-sol01_0 - 2007 01 " 20106 01 " 1 . lim...

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Unformatted text preview: 2007 01 " 20106 01 " 1 . lim sin ax = 1 , a 0 : .1 x 0 ax , y 0 x 0 . y = ax sin y sin ax = lim =1 x 0 ax y 0 y lim : : sin 5 x sin 5 x - sin 3 x sin 5 x - sin 3 x x sin 3 x x lim = lim = lim 5 - 3 = x 0 x0 x sin x sin x x 0 5x 3 x sin x 1 1 1 x sin 5 x sin 3 x = 5 lim - 3 lim = ( 5 - 3) 1 = 2 lim x0 x0 x 0 3x 5x sin x . - . sin x = - sin x , sin x < 0 - < x < 0 .2 , lim- x0 sin x - sin x sin x = - lim- = - 1 : x0 x0 x x x (! 1 1 0 - ) . = lim- 2 . x 2 + x3 1 + x 1+ x = = x3 + x 6 x + x 4 x (1 + x3 ) .1 , 0 - -x . 0- - , 0 - x + x3 1 + x2 1 + x2 = lim = lim x 0 x3 + x6 x 0 x 2 + x5 x 0 x 2 (1 + x3 ) lim x0 x0 .2 lim (1 + x 2 ) = 1 0 ,0 , lim x 2 (1 + x3 ) = 0+ . . 1 2007 01 " 20106 3 . x 2 cos 1 x tan x = x 2 cos 1 x sin x cos x = x 2 cos 1 cos x x sin x = x x cos 1 cos x x sin x .1 ' , - 1 cos 1 cos x 1 x 0 x . lim x cos 1 cos x = 0 x x0 .( I .73 ") lim x 2 cos 1 x tan x x =1 x 0 sin x . lim x0 = 1 0 = 0 3 2 1 + + 2 + +1 x5 x7 = 0 + 0 + 0 = 0 lim = lim x 1 x x5 + x 7 + 1 x 1 0 +1+ 0 +1+ x2 x7 3 x5 2 x2 .2 4 : x > 0 .1 x2 + x - x = = ( x 2 + x - x )( x 2 + x + x ) x2 x2 +x +x 1 = A +x +x x : , x = A= 1 = x2 + x +1 x2 = x2 x = +x +x x 2 , x > 0 1 = x2 + x +1 x2 1 1+ 1 x +1 x lim x 2 + x - x = lim 1 1+ 1 x 1 y x +1 = 1 1 = 1+1 2 .2 x - lim ( x 2 + x - x) = lim( y 2 - y + y ) = lim( y 1 - y y 1 + y ) = . y = - x . 2 2007 01 " 20106 5 : , .1 - x 2 . . f ( x ) = -1 x0 x=0 1 g ( x ) = x2 0 x 0 x =0 x0 = 0 . .2 . lim f ( x ) = f ( x0 ) < 0 x0 - f - x x0 . - ( ' I .84 ) 6 . y = x0 + 1 . f ( x0 ) 0 - x0 0 . . f ( y ) < 0 , f ( y ) < f ( x0 ) 0 , f x > N N > 0 , lim f ( x ) = 0 . = f ( y ) > 0 x .1 (*) f ( x ) < = f ( y ) f ( x ) < f ( y ) < 0 f x > y . x > max{ N , y} . f ( x ) > f ( y ) . , f ( x ) < f ( y ) - (*)- , x > N . x [0, ) f ( x ) > 0 , : : , .2 1 . f ( x) = 2 x 1 x x=0 xN ' , lim 0 = lim x 2 x x = 0 , 0 < f ( x) 2 x > 0 x x . f , [0, ) - f - . lim f ( x) = 0 - . x (n, n + 1) .( n = [ M ] + 1 - ) n > M , M > 0 . [ M , ) - f , f ( x) = 1 1 2 < = f (n + 1) , x < n + 1 x n n +1 3 2007 01 " 20106 7 f (a ) . .1 L a . f (a) = L . f - . f (a) > L * x N (a ) , > 0 lim f ( x) = L . = f (a) - L x a . f ( x) < f (a ) f ( x) - L < f (a) - L , f ( x) - L < . f - , f ( x) < f (a ) " , a < x < a + . f (a) > L / .a- f- f (a) = L , f (a) < L - . f- . lim g ( x) = - L - (?) g . g ( x) = - f ( x) x a . g (a) = - L ,a- g , g .a- f , f (a) = -(- L) = L g I .60 - . .2 . L = 0 , L = lim+ f ( x ) 0 x a L = lim- f ( x ) 0 x a 8 , lim f ( x) = f ( x0 ) > g ( x0 ) = lim g ( x) x0 - . x x0 x x0 .1 . I .71 . x0 = 0 f ( x ) = x 2 : . .2 9 ,( I .84 ) : . . . .1 x > M x M > 0 . N > 0 . f ( x) - g ( x) < - N 4 2007 01 " 20106 x > M1 x M1 > 0 , lim f ( x ) = 0 x . f ( x) < 1 , f ( x) < 1 x > M 2 x M 2 > 0 , lim g ( x ) = x . - g ( x) < - N - 1 , g ( x) > N + 1 x > M1 , M 2 x > M . M = max{M1 , M 2 } . , f ( x) - g ( x) < 1 - N - 1 = - N . .2 . f ( x) g ( x) < x > M M > 0 . > 0 . f ( x) < 1 x > M1 M1 > 0 , lim f ( x ) = 0 x x x > M 2 M 2 > 0 , lim g ( x ) = .0 < 1 . N > 1 1 < , g ( x) > N > g ( x) x > M1 , M 2 x > M . M = max{M1 , M 2 } . , f ( x) f ( x) = < 1 = g ( x) g ( x) 10 : , .1 . . lim g ( x) = x f ( x) = sin x x sin x 1 sin x 1 x > 0 = 0 ' , - x x x x x . lim g ( x ) = lim x = - x x f ( x ), g ( x ) > 0 (2 k , 2 k + ) , (2 k + , 2 k + 2 ) ,( ) - . g ( x) f ( x) , x = n f ( x ) = 0 - : .( ) - , ( a , ) 5 2007 01 " 20106 : 1 , .2 ( f g )( x ) = f ( x ) g ( x ) = x sin x = sin x x > 0 x x .( 1 - - 1 ) lim sin x - 11 : - .( ) x - 1 < [ x ] x : x 2 2 2 : .1 -1 < x x x . 2 x x 2 x 2 x 2 2 - = - 1 < = x 3 x 3 3 3 3 x 3 , : x > 0 2 x x 2 x 2 x 2 2 : x < 0 - = - 1 > = x 3 x 3 3 3 3 x 3 (- ) ' , lim . lim x 2 x 2 2 = lim = 3 x x 0- 3 x 3 x =0 x 0 3 x 0+ . .2 . lim x 3+ x 3 x x = lim 0 = 0 . = 0 0 < < 1 x > 3 3 3 x x 3+ 3 12 . = x 1 1 . .1 10 ,4 , lim f ( x ) 1 . .2 .1 ( I .57 ) 13 ,4 , lim f ( x ) . .1 x 1 . ( I .57 ) 6 2007 01 " 20106 1 .2 .( ) .4 .4 14 . f (0) = 0 , f (0) = f (0 + 0) = f (0) + f (0) : . .1 , 0 = f (0) = f ( x - x ) = f ( x + ( - x )) = f ( x ) + f ( - x ) . - f (- x) = f ( x) > 0 0 - . > 0 , x0 R . .2 .(*) f ( x ) - f (0) = f ( x ) < x < =0 . f ( x - x0 ) < (*)- x - x0 < , : . > f ( x - x0 ) = f ( x + ( - x0 )) = f ( x ) + f ( - x0 ) = f ( x ) - f ( x0 ) . x0 - f 15 .(3 66 ) D ( x ) - . .1 - ' , D ( x ) . g ( x ) = x D ( x ) .0 - g , lim xD ( x ) = 0 = g (0) x0 . x0 - g ( x ) - x0 0 , x0 0 - x- g ( x ) - , D ( x ) = g ( x) x x0 , x0 - . .0 - g ( x ) . , 1 .2 7 2007 01 " 20106 16 : , .1 1 x0 . f ( x) = 0 , g ( x) = , a=0 - 1 x < 0 ,0 - f g , ( f g )( x) = 0 x .0 - g- 0 - f- . .2 .a- f - ,a- g , g ( x) = ( f + g )( x) - f ( x) : a . . f g . : 8 ...
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This note was uploaded on 07/23/2009 for the course MATH. 04101 taught by Professor ישראלפרידמן during the Summer '06 term at The Open University.

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