1. Математический анализ (850924), страница 7
Текст из файла (страница 7)
12.4. 9, ) A 1 ( )!)! D#, % %% % $ , 1(g(x) x.11300000000 (!' '!". f (x) g(x) ( f (x) g (x) 6= 0 x0, o &- x0..xlima f (x) = xlima g (x) = 0 , fg ((xx)) x ! x0 , f (x)f (x)(12:4)xlimx g (x) = xlimx g (x) :00!!000!0!00%. " +, ) %$ $)+, !, <%=! f (x) g(x) )% x = x0. 6 'F+++, ) # <%= ! )% '$$), ! )% , , '$$) %1, ) $)+ , <%=! !)%.
$ f (x) g(x) )%x = x0 , 1 f (x0 ) = g(x0) = 0. 6 + ' #. ; $ (xlima f (x) = f (x0 ) xlimx g (x) = g (x0 )!!0.. <%= f (x) g(x) )% x = x0. $% x0 x], x > x0, $% x x0], x < x0, x{ '+ )% %! % )% x0, <%= f (x) g(x) + + D#.8 )! $% x0 x], x > x0. D# *, %!! , %+ )% c, ) f (x) = f (x) ; f (x0 ) = f (c) :(12:5)g(x) g(x) ; g(x0) g (c). %, )% ' '(# !, %) )% c (12.5) $( %%-' ( $ ,. )% c ' <%=! , *! $) + %1 x,) c ! x0 + 0, % x ! x0 + 0.11400 x ! x0 # fg ((xx)) .6 $ ' + x % )% x0. x ! x0 +0, c ! x0 +0, # fg ((cc)) , *! #+ fg ((xx)) :f (c) = lim f (x) :limc x +0 g (c) x x +0 g (x); $ (12.5) ,)f (x) = lim f (x) :(12:6)limx x +0 g (x) x x +0 g (x)I ) %$+ f (x) = lim f (x) :lim(12:7)x x 0 g (x) x x 0 g (x)8 (12.6) (12.7) $+ 1 ( ( (12.4).
%$. 12.1.1 ;12 x)1(1;costgx;x2xcos= lim= xlim0 cos2 x(1 ; cos x) =limx 0 x ; sin x x 0 1 ; cos x00000000!00!000!00!0!0;0!0;0!!!cos x = 2:= xlim0 1 +cos2 x 12.5. ? (, ) # $ , , * o1 1( +1 +, ) # <%=! . ;,f (x) = x2 sin x1 g(x) = x:!f (x) = lim x sin 1 = 0limx 0 g (x) x 0x!!115f (x) = 2x sin 1 ; cos 1g (x)xx2 x ! 0. 12.6. )f (x)limx 0 g (x) , + +( A+ %(% $. %, + + (% <%= f (x) g(x), , $ f (x) g (x), + )+f (x)limx 0 g (x)1 +( ($(+ A+. 6 )f (x)limx 0 g (x)% )f (x) : :limx 0 g (x) 12.2.x ; sin x = lim 1 ; cos x = lim sin x = lim cos x = 1 :limx 0x 0 6xx 0 6x 0x33x26 A+ % ++ )!, % x + % '%). ! 12.2. f (x) g(x) b +1),lim f (x) = x limg(x) = 0x ++( f (x) g (x) 6= 0 (b +1).
.f (x) limx + g (x) f (x)limx + g (x)00!0000!!0000!!!!!1!000!!1011161! f (x) = lim f (x) :limx + g (x) x + g (x)0!%.y ! +0, 1!f (x)limx + g (x)!101 x = y1 . , x ! +1, 1!1!lim f y = xlim+0 g y = 0:x +0!!1 f y1= ylim+0 1 g y!= ylim+0!fg 00 1 ;12y y 11y ; y2=f 1yf (x) := ylim+0 1 = x lim+ g (x)g y %$.I ) + A+ x ! ;1.0!0!010 ( ) A+ 1 '( ($ ) *, :1)xlimx0!f (x) g(x) xlimx f (x) = 0 xlimx g(x) = 1:!0!0@+ ! = ( 0 1 + % 00 11 , % %! $1 ( A+:f (x) g(x) = f (1x)g(x)f (x) g(x) = g(1x) :f (x)117 12.3.11) xlim+0 x ln x = xlim+0 ln1x = xlim+0 x1 = ; xlim+0 x = 0:; x2xx = lim x = lim 1 = 0:2) x limxe+x + ex x + ex3)xlimx (f (x) ; g (x)) xlimx f (x) = 1 xlimx g (x) = 1:!!!!;!!10!1!0!1!0; ( (1 ;1) % + % 00 :1 ; 1f (x) ; g(x) = 11 ; 11 = g(x) 1 f (x) f (x) g(x)f (x)g(x)g(x) xlimx0 f (x)4)! + $ , ):) xlimx f (x) = 0 xlimx g(x) = 0!0!0')xlimx0f (x) = 1 xlimx g(x) = 1)xlimx0f (x) = 1 xlimx g(x) = 0:!!!0!0" ) + 1y = f (x)g(x) ( y = eg(x) ln f (x) ($(+ A+ ) xlimx0!g(x) ln f (x):118 12.4.!1) xlim+0 xsin x = exlim+0sin x ln x!% %%= e0 = 11lnxlim sin x ln x = xlim+0 x ln x = xlim+0 1 = xlim+0 x1 =x +0;xx= xlim+0 (;x) = 0:!!!!!! ctg2) xlim0 (cos x)ctg x = exlim2% %%20x ln cos x!=e 12;2 x ln cos x = lim ln cos x lim cos2 x = lim ln cos x =limctgx 0x 0x 0 sin2 x x 0x2!!!!; sin x = ; 1= xlim0 cosx 2x2!(, )lim cos x = 1x 0!sin x x ! 0).
1. 0 "+ 6.p32. 7 f (x) = 1 ; x !"1" x = ;1 x = 1, f (x) 6= 0 " " (;1 1). & "" " 6 f (x)" 8;1 1]? (c . " " 12.1).3. & f (x) = x(x ; 1)(x ; 2)(x ; 3). &"+, " f (x) = 0 ; .4. 0 "+ "'"+".5. & "'"+" 21200f (x) = 1 + x + x2" 80 2]. <" (1 " C .1196. & "'"+", "+ " j sin x ; sin yj jx ; yj( . "+ 12.2).p7. =+ "'"+" y = 1 + 3 x " 8;1 1]?8. 0 "+ >?.9. @"+, f (x) a " 8a b] f (x) = 0, 8x 2 (a b) f (x) const 8x 2 8a b]:10.
0 "+ " ".11. =+ " " ";+( "x sin x ?limxln(1 + x) " (c . " " 12.4).2012!0p2. > f (x) = 1 ; 3 x " 8;1 1] 6," ", ; f (x) " " 8;1 1], " 1 x = 0, .. 2) .3. & f (x) 6 " 80 1]. 3" " f (x) = x(x ; 1)(x ; 2)(x ; 3) " ( ( ;" x, 6 !; 3) : f (0) = 0 f (1) = 0. < '" 6, 1 "" "C 2 (0 1) f (C ) = 0:A"', 6 "? " 81 2], ae , 1 "" " C 2 (1 2) f (C ) = 0: & x(x ; 1)(x ; 2)(x ; 3) 6 " 82 3] (" , f (2) = f (3) = 0),ae 1" " C 2 (2 3), f (C ) = 0.3" !" , "" 1" ; C C C , ;f (x) = 0:5. 7 f (x) = 1 + x + x " " " ( ( f (x) = 3x + 1.
0", " " 80 2] "'"+" f (2) ; f (0) = 11 ; 1 = f (c) C 2 (0 2)2;02f (C ) = 5 3 C + 1 = 5 3 C = 4 C = 43 C = p2 :3p37. <, " " y = 1 + x ; " " 8;1 1], ( x = 0.9. ( x 2 (a b] "'"+" f (x) " 8a x ]:f (x ) ; f (a) = f (C ) = 0:x ;aC", f (x ) = f (a) 8x 2 (a b] " , f (x) const:102102203310320020222000000012023 13+ , ! 13.1 (/!). f (x) a b] (n ; 1)-# , (a b) n- f (x).
"# , - (a b), (n 1) (a)f(a)ff(a)2f (b) = f (a) + 1! (b ; a) + 2! (b ; a) + : : : + (n ; 1)! (b ; a)n 1+(n) ( )f(13:1)+ n! (b ; a)n:%. 9'$) )$ Rn $( 1 $)+ <%= f (b) 1+(n 1) (a)f(a)f(a)f2f (a) + 1! (b ; a) + 2! (b ; a) + ::: + (n ; 1)! (b ; a)n 1:2Rn = f (b) ; 4f (a) + f 1!(a) (b ; a) + f 2!(a) (b ; a)2 + : : : +3(n 1) (a)fn1+ (n ; 1)! (b ; a) 5 :; f (b) = f (a) + f 1!(a) (b ; a) + f 2!(a) (b ; a)2 + : : : +(n 1) (a)f(13:2)+ (n ; 1)! (b ; a)n 1 + Rn:L %( % Rn M (b ; a)n (o) , ) % ) *: = (b ;Rna)n ).000;;000;;000;;000;;f (b) = f (a) + f 1!(a) (b ; a) + f 2!(a) (b ; a)2 + : : : +(n 1) (a)f+ (n ; 1)! (b ; a)n 1 + M (b ; a)n:121000;;(13:3)8 ( <%='(x) = f (x) + f 1!(x) (b ; x) + f 2!(x) (b ; x)2 + : : : +(n 1) (x)f+ (n ; 1)! (b ; x)n 1 + M (b ; x)n($, ) <%=+ '(x) % )+, ! ) (13.3) ( ).M%=+ '(x) + $% a b] + 8+.
@!(, !, , <%=f (x) ! !, :1) '(x) $% a b]2) * $ + ' (x), %!! , (a b), )' (x) = f (x) + f (x)(b ; x) ; f (x) + f 2!(x) (b ; x)2 ; 2f (x)(2!b ; x) + : : : +(n) (x)f1 f (n 1) (x)(b ; x)n 2 ; Mn(b ; x)n 1 =+ (n ; 1)! (b ; x)n 1 ; (nn ;; 1)!(n)= (fn ;(x1)!) (b ; x)n 1 ; Mn(b ; x)n 1:(13:4)3) %=, $% a b] <%=+ '(x) % $)+(n 1) (a)'(a) = f (a) + f 1!(a) (b ; a) + : : : + f(n ; 1)!(b ; a)n 1 + M (b ; a)n = f (b)(. < (13.3))'(b) = f (b):? (, 8+ (a b) *, %!! , %+ )% , ) ' ( ) = 0: % '$,000;;00000000000;;;;;0;;;0(n) ( )f' ( ) = (n ; 1)! (b ; )n 1 ; Mn((b ; )n 1 = 0:0;;122(13:5)?%*+ < (13.5) 1 n(b ; )n 2 6= 0 ( 2 (a b)),)ae(n) ( )(n) ( )ffM = n(n ; 1)! = n!, (,n 1) )(n)f (b) = f (a) + f 1!(a) (b ; a) + ::: + f(n ; (1)!(b ; a)n 1 + f n(! ) (b ; a)n: %$.(n) ( )f 13.1.
"1 Rn = n! (b ; a)n $+ ) ) < ! < A 1, <(13.2) $ " /#-. 13.2. 1 < (13.2) n 1.f (b) = f (a) + f 1!( ) (b ; a):+ f (a) )( $ ) 1 (b ; a), )ae < A 1f (b) ; f (a) = f ( ) 2 (a b):(b ; a)6 $), ) ! +++ ''* A 1. 13.3.
" (13.2) a b 1 $+( $( )% x0 x $ $% a b]. < ! +<%= f (x) 1 $( * :(x0) (x ; x )2 + : : : +f (x) = f (x0 ) + f (1!x0) (x ; x0) + f 2!0(n 1) (x0 )(n) ( )ffn1(13:6)+ (n ; 1)! (x ; x0) + n! (x ; x0)n )% 1 1 )% x x0 = x0 + (x ; x0 ), 0 < < 1.1 < (13.6) x0 = 0 (n 1) (0)(n) ( )fff(0)2n1f (x) = f (0) + f (0)x + 2! x + : : : + (n ; 1)! x + n! xn: (13:7)123;0;;00000;;000;;M (13.7) $ < 5% )) < A 1. 9)( ) ($ < ! ', $ %% $)+ ) ). "1 $(#( ) ) % 0, ,$( e + % .
! 13.2 ((!). f (x) n-# ( U (x0) x0. "# (& 0 " .(n)f (x) = f (x0 ) + f (1!x0) (x ; x0) + ::: + f n(!x0 ) (x ; x0)n + o((x ; x0)n):(13:8)%. ! + , 2 U (x0) < ! ) ) < A 1 (13.6).(n) ( )ff(x0)(13:9)f (x) = f (x0 ) + 1! (x ; x0) + ::: + n! (x ; x0)n: f (n)(x) )% x0 , (,00f (n) ( ) = f (n) (x0) + f (n)( ) ; f (n) (x0)] = f (n) (x0) + (x) (x) = f (n) ( ) ; f (n) (x0) + % x ! x0, % %% )% 1 1 )% x x0 x ! x0 )% ! x0 , f (n) (x) )% 0, f (n) ( ) ! f (n) (x0). T% '$,f (n) ( ) = f (n) (x0) + (x) xlimx (x) = 0!0 < (13.9) $+ (n) (x0 )ff(x0)f (x) = f (x0 ) + 1! (x ; x0) + ::: + n! (x ; x0)n + n(x! ) (x ; x0)n:1240% %% xlimx (x) = 0, n(x! ) (x ; x0)n = o((x ; x0)n) !0(n) (x )f(xf0)f (x) = f (x0 ) + 1! (x ; x0) + : : : + n! 0 (x ; x0)n + o((x ; x0)n):(13:10) %$.
13.4. M (13.10) $ " , 1 Rn = o((x ; x0)n) < (13.10) $ ) ) < . 13.5. . < (13.10) 1( 0 = 0, )<(n) (0)ff(0)(13:11)f (x) = f (0) + 1! x + ::: + n! xn + o(xn)%+ $ 1 .00 ex sin x cos x ln(1 + x) (1 + x)) ( f (x) = ex. % %% (ex)(k) = ex k = 1 2 : : :, f (0) = 1 f (k) (0) = 1 , (, 3n2ex = 1 + x + x2! + x3! + ::: + xn! + rn(x) rn { )! ) < !: < A 1xern = (n + 1)! xn+1 < rn = o(xn )') ( f (x) = sin x, sin(k) x = sin(x + k=2) k = 1 2 : : : (, f (0) = 0 f (2m) (0) = sin m = 0 f (2m 1) (0) = sin(m ;;=2) = sin((m ; 1) + =2) = cos((m ; 1)) = (;1)m 1 :;;125,3 x52m 1xxm1sin x = x ; 3! + 5! ; : : : + (;1) (2m ; 1)! + r2m(x) r2m { )! ) < !: < A 1+ =2) x2m+1r2m = sin(x(2+mm+ 1)! < r2m = o(x2m )(, ) $1 sin x < 5% '+n = 2m ) f (2m) (0) = 0).) I ), ( f (x) = cos x.
T , cos(k) x = cos(x + k=2)k = 1 2 : : : (, f (0) = 1 f (2m) (0) = cos m = (;1)m f (2m 1) (0) = cos(m ; =2) = 0:2 x42mxxmcos x = 1 ; 2! + 4! ; : : : + (;1) (2m)! + r2m+1 (x) r2m+1 { )! ) < !: < A 1+ (m + 1)) x2m+2r2m+1 = cos(x(2m+ 2)! < r2m+1 = o(x2m+1 )(, ) $1 cos x < 5% '+n = 2m + 1). ) 8 ( <%= f (x) = ln(1 + x). % %%(ln(1 + x)) = 1 +1 x ;;;0(ln(1 + x))(k)k 1 (k ; 1)!(;1)= (1 + x)k k = 2 3 : : :;126, f (0) = 1 f (0) = 1 f (k) (0) = (;1)k 1 (k ; 1)! k = 2 3 : : :, , (, )+, ) (k ;k! 1)! = k10;3n2ln(1 + x) = x ; x2 + x3 ; : : : + (;1)n 1 xn + rn(x) rn { )! ) < !: < A 1n(;1)rn = (1 + x)n+1(n + 1) xn+1 < rn = o(xn )) ( f (x) = (1 + x) ;((1 + x) )(k) = ( ; 1) : : : ( ; k + 1)(1 + x)( k) k = 1 2 : : :; f (0) = 1 f (k) (0) = ( ; 1) : : : ( ; k + 1) k = 1 2 : : :, (1 + x) = 1 + x + (2!; 1) x2 + : : : + ( ; 1):::n(! ; n + 1) xn + rn(x) rn { )! ) < !: < A 1: : : ( ; n) (1 + x) n 1 xn+1rn = ( ;(1)n + 1)! < rn = o(xn ):;; 1) L ;( ( = m { ( )).(1 + x)m = 1 + mx + m(m2!; 1) x2 + : : : + m(m ; 1) : :k:!(m ; k + 1) xk +: : : 1 xm + : : : + m(m ;m1)!127(1 + x)m = 1 + mx + m(m2!; 1) x2 + ::: + mxm 1 + xm :2) = ;11 = 1 ; x + x2 ; x3 + ::: + (;1)n xn + o(xn )1+x1 = 1 + x + x2 + x3 + ::: + xn + o(xn ):1;x ;M ! $+ ' <%=, +* + ! % %! )%,$( ) )( '%) , ' % + %, ) ) ).
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