infi-07b-sol02_0 - 2007 02 " 20106 02 " ,...

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Unformatted text preview: 2007 02 " 20106 02 " , : . 1 .( 4 2 ") .1 . (0,1) : . .2 2 .0 - , lim x sin 1 = 0 .1 x x0 (?) R - f ( x ) , f ( x ) = x sin 1 x 0 x0 x=0 x lim f ( x ) = lim x sin x 1 x y= 1 x = lim+ y 0 y= 1 x sin y y =1 x - lim f ( x ) = lim x sin 1 = lim x x - 1 2 sin y y y 0- =1 , 1 < f ( x) < 2 3 2 , f ( x ) - 1 < 3 2 x > N N > 0 , 1 2 . 1 < f ( x) < 2 , f ( x ) - 1 < x < - M M > 0 ,( ) , f [ - M , N ] . f ( x ) < K x K > 0 x R\{0} , . f ( x ) < L x R , L = max{K , 3 } 2 . A , x sin 1 < L x N > 0 , B , lim x cos 1 = x x 1 .2 . , . x cos > N - x 1 x 3 . f ( x) f (sup A) , x sup A x A . .1 . f ( A) f (sup A) . sup( f ( A)) f (sup A) , f ( A) sup( f ( A)) 1 2007 02 " 20106 : , .2 . A = {- 1}, B = {1} , f ( x) = x 2 x0 x>0 2 x sup ( f ( A) ) + sup ( f ( B ) ) = - 1 + 2 = 3 > sup ( f ( A + B ) ) = 0 2 2 4 . 10 5 . .1 . - y R . x0 - f - . .2 .( y < x0 ) y > x0 - . f ( y ) > f ( x0 ) x ( x0 , x0 + ) > 0 , x0 . x0 < x < y , x ( x0 , x0 + ) ( x, y ) . f ( x ) f ( x0 ) . f ( x ) < f ( x0 ) - f ( x ) f ( x0 ) , f ( x ) f ( x0 ) f ( x ) < f ( x0 ) < f ( y ) . [ x, y ] - , R - f ( x0 < x < z ) z x0 . f ( z ) = f ( x0 ) - z ( x, y ) . . x0 - f- , : 6 . x0 = 0 , f ( x) = sin 1 x -2 x0 x=0 : .1 x N ( x0 ) 1 > 0 , x0 - . .2 1 .(1) f ( x) f ( x0 ) x N ( x0 ) 2 > 0 , x0 - 2 .(2) f ( x) f ( x0 ) , f ( x) f ( x0 ) f ( x) f ( x0 ) x N ( x0 ) . = min{1 , 2 } . f ( x) = f ( x0 ) 2 2007 02 " 20106 7 . [0,1] f ( x ) = 1 g ( x ) = x 0 x < 1 : x =1 0 .1 . sup f (( a , b)) = sup g (( a, b)) = 1 , [0,1] - f ( x ) > g ( x ) , (0,1) - g - f .( ) [ a , b ] - g . .2 . x0 - . sup g ([ a , b]) = max g ([ a, b ]) = g ( x0 ) < f ( x0 ) sup f ([ a, b ]) : 8 - - 0 - . (- 1,1) , f ( x) = sin 1 x 0 x0 x=0 : .1 ! . (0,1) f ( x) = sin 1 , , .2 x 9 : , III .14 III.14* , ( a , b ) f . lim- f ( x ) x b .1 . lim+ f ( x ) , f xa f ( x ) , III .14 - f ( x ) () f ( x ) , - f ( x ) .() . .f - x0 ,(?) f ( x0 ) - (-, x0 ) - f . lim f ( x) III .14 * - x x0 ,(?) f ( x0 ) - ( x0 , ) - f . lim f ( x) III .14 * x x + 0 3 2007 02 " 20106 - (! ) x0 - - . - . . - f - x0 - . f ( x0 ) L - , lim f ( x) = L - L R x x 0 .( f ( x0 ) > L ) f ( x0 ) < L . f ( x) > f ( x0 ) x < x0 , f - , lim f ( x ) = L f ( x0 ) (- ) ' I .71 , - x x0 . f ( x0 ) < L . - x0 - f - . - : f I . f ( x ) = x . .2 . f , 10 . g ( x1 ) = a g ( x2 ) = b - x1 , x2 [ a , b ] , g (x) .1 . a f ( x ) b x f . h( x) = f ( x) - g ( x) . h( x2 ) = f ( x2 ) - b b - b = 0 h( x1 ) = f ( x1 ) - a a - a = 0 . ( [ x 2 , x1 ] - ) [ x1 , x 2 ] - h - ( x0 [ x 2 , x1 ] ) x0 [ x1 , x 2 ] . f ( x0 ) = g ( x0 ) , h( x0 ) = 0 . [ a , b ] = [ - 1,1] - g ( x) = x ,1 . .2 11 . > 0 .1 .( ) [0,4 ] sin x x - y < 1 x, y [0,4 ] 1 > 0 (1) . sin x - sin y < . x y . x - y < x, y R = min{ 1 , 2 } .(? n ) 2n x < 2n + 2 - , n 4 2007 02 " 20106 2n y < 2n + 2 + 2 = 2n + 4 0 y - x < 2 - : . x1 = x - 2n y1 = y - 2n (2 ) (3 ) (4 ) sin x = sin x1 sin y = sin y1 x1 - y1 = x + 2n - ( y + 2n) = x - y < 0 x1 < 2 0 y1 < 4 ((3)-) x1 - y1 = x - y < 1 ((4)-) x1 , y1 [0,4 ] . , sin x - sin y < (2)- sin x1 - sin y1 < (1)- . .2 ,0 < 1 1 < . n > n > 1 . > 0 , = 1 n . , 0 < x1 , x2 < x1 = 1 2 n + 2 1 < 1 n , x2 = 1 2 n + 3 2 , x1 - x2 < - . x1 , x2 (0,1) - . sin 1 1 3 ) = 1 - ( - 1) = 2 > - sin = sin(2 n + ) - sin(2 n + x1 x2 2 2 . (0,1) - sin 1 x 12 .L " " . .1 : x - y < x, y R . = L , > 0 . , f ( x) - f ( y ) L x - y < L L = .2 : x, y I (*) f ( y ) - f ( x) f ( y ) - f ( x) 1 1 - = f ( x) f ( y) f ( y) f ( x) L2 x - y < x, y I > 0 I- " f - . > 0 . f ( x ) - f ( y ) < L 2 f ( x) - f ( y) L 2 1 1 - < = f ( x) f ( y) L2 L2 (*)- x - y < 5 2007 02 " 20106 . I - 1 f ? . L 0 : 13 .11 , (0,1) f ( x) = sin 1 : .1 x . f - . f ( x ) = sin( x 2 ) : . .2 ( ) . I ,J- , R - f . I J - J f ,5 23 - , J- f- .I- ) R - , f , .( 13 14 " . .1 : ." . , , ," . .5 23 .2 15 (8 III .14 * -) III .14 . .1 . , lim+ f ( x) x a x b - lim f ( x) x=a lim+ f ( x) xa 0 < x <1 g ( x) = f ( x) lim f ( x) x=b x b- : . (a, b) - " g ( ) " , [a, b] - g ( x) . (a, b) - f , g ( x) = f ( x) (a, b) - . b = ( ) III .14 . .2 : 6 2007 02 " 20106 . lim f ( x) (a, ) - f x (-, b) - f . lim f ( x) x - . , , 12 , . 7 ...
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