1. Математический анализ (850924), страница 10
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, (! ++ <%= +1, ) , ($+ 1 , ) ) (!. 2.2. ")(4;1x:limx + 3x4 + 1161!1!14;14x1;1=x1 ; 0 = 1: >< x lim=lim=44+ 3x + 1 x + 3 + 1=x3+0 3. x limf (x) = 0 <%=+ f (x) $+ & + x ! +1.M%=+ f (x) $+ & &, x ! +1, 8M > 0 9N : 8x > N jf (x)j > M ) #lim f (x) = 1:x +.f (x) = 1limx + g (x) <%= f (x) g(x) $+ % x ! +1,..f (x) g(x) x ! +1:51 <( $ <%=! % ) , ) + (! (. < (1.2)).!!1!11!1!1 ( 2.3. ;!3 ; x22x:limx + x4 + 3x3 + 13 ; x232x2x2 = 0: >< x lim=lim=lim434+ x + 3x + 1 x + xx + x!!1!11!1 2.4.
;!41+x+6xlim 1 + 2x3 :x +441+x+6x6x< x lim= lim 3x = 1: >++ 2x3 x +1 + 2x3 = x lim!!1!11 2.5. ;!!12 ; 5x3xlim:x + 2x2 ; x + 72 ; 5x23x3x3: >< x lim=lim=+ 2x2 ; x + 7 x + 2x22162!!1!11% 1 %% + (! +, ) Pm (x) Pk (x) { ) ! m k , 8> 0 m < kPm (x) <lim= > a m = kx + Pk (x): 1 m > k,!1 a { # %<<= #, +, )(. 1.9{1.11 < (1.3)). 2.6.
")(p2x ; 1:limx +xp< "+ x2 ${ %+ )+, ) x2 = jxj = x x > 0, )!limpvqux2 ; 1 = lim x 1 ; 1=x2 = lim ut1 ; 1 = 1: >x +x +xxx2x +!1!11!1(' 2" ! "$ &'"<" (1 :2) x lim x ;+x 2+xx+ 1 1) x lim (xx +;1)1 4!+ 1233!+13) x lim (x ; 1) ;x (x ; 3) :22!+11) 1 2) 0 3) 4:||||{( <%=+ f (x) x < x0. Q a $+ <%= f (x) ;1 ..lim f (x) = a 8" > 0 9N > 0 : 8x < ;N jf (x) ; aj < ":x!;1 )+ x lim f (x) ) )+lim f (x). ; <%=+ $ + 1, 1* x +% ), !, 1 ++( 1(.163!;1!1 2.7. ;!p2;1xlimxx :< :)+, ) jxj = ;x x < 0, )!;1pqx2 ; 1 = lim jxj 1 ; 1=x2 =limxxxxvqu2ux1;1=xt1 ; 1 = ;1: >= x lim ;=lim;xxx28 .
N$ 2.6, 2.7 , )!;1!;1!;1!;1p2x ; 1 x x ! +1 p2x ; 1 ;x x ! ;1:(' 2" ! "$ &'"<" (1 :p1) x lim p3 x + 1 x +1p43) x lim 16x +x 2x + 3 p2) x lim px + 1 x +1p2!+1234!;124) x lim!+11) 12) ;13) 2!;134316x + 2x + 3 :x424) ;2:||||{( <%=+ f (x) jxj > x0 > 0, .. x > x0 x < ;x0 .Q a $+ <%= f (x) 1 #xlim!1f (x) = a 8" > 0 9N > 0 : 8jxj > N jf (x) ; aj < ":9) , xlim f (x) = a (% , % !1lim f (x) = a x lim f (x) = a ::x +!1!;1164f (x) = a () x limf (x) = a = x lim f (x):+.
x limf (x) 6= x lim f (x) ,+ ' $ , *+, xlim f (x) * (9).N$ $(, ), , 2.6, 2.7, , )p29 xlim x x; 1 : 2.8. ;!xlim!1!!11!;1!;1!1!1xlimpx2 + 1 ;!1px2 ; 2:< :1 $ 1 +1 $ $( % ).xlimpx2 + 1 ;!1= xlimpx2 ; 2=p 2 pppx + 1 ; x2 ; 2 x2 + 1 + x2 ; 2ppx2 + 1 + x2 ; 22 + 1) ; (x2 ; 2)(xp 2 = xlim 2j3xj = 0: >p= xlim 2x +1+ x ;2 2.9. ;!!1=!1!1xlim!1xpx2 + 1 ; x:< 8 ( x ! +1 x ! ;1. ) x ! +1 ' !(%1 %% * :p2p2px+1;x)(x + 1 + x) =x(2 + 1 ; x = limpxlimxx +x +x2 + 1 + xx = 1:= x lim+ 2x 2")p2lim x x + 1 ; x :x!!1!11!;1165p% %% x ! ;1 <%= x2 + 1 (;x) +++'(# ' 1(, $(p 2 '%)x + 1 ; x ' '%) '(#! x ! ;1 xlim xp!;1N%,px2 + 1 ; xx2 + 1 ; x= ;1:p12= 2 6= x lim x x + 1 ; x = ;1:lim xC (,p2xlim x x + 1 ; x *: > 2.10.
;!qq22(x + 1) ; (x ; 1) :xlim< @+ )+ ($ 1+ + +1 1:qq22(x + 1) ; (x ; 1) =xlim2 ; (x ; 1)2(x+1)= xlim q qq =(x + 1)2 2 + (x + 1)2(x ; 1)2 + (x ; 1)2 22 + 2x + 1 ; x2 + 2x ; 1x= xlim q qq =(x + 1)2 2 + (x + 1)2(x ; 1)2 + (x ; 1)2 2= xlim 3x4x4=3 = 0: >x +!1!;1!133!133!1!1333!1333!1(' 2" ! "$ &'"<" :pp1) xlim x x + 3 ; x ; 2 pp3) xlim x 3 5 + x ; 3 2 + x :44!123q2) xlim x ; x(x ; 1)!13!11) 02) 13) 1:||||{166 3( + ) '" <%= '%) ($ <%=!, , '%), , (a +1) (;1 b) (;1 +1).
@+ )+ + + %, <%=! $ )1 1 ($( .++y = kx + b(3:1)$+ <% <%= y = f (x), (x) = f (x) ; (kx + b) ! 0 x ! 1.., (3:2)(x) = 0:. (3.1) *, (3:3)k = xlim f (xx) b = xlim f (x) ; kx]:. ,+ ' $ (3.3) *, ' 1, <% <%= y = f (x) (3.1). k 6= 0 $+ , k = 0 { #.. (x) ! 0 (% x ! +1 ( (% x ! ;1), $+ ( ). + (3.2) ++ (3.1) +++ ! ( !, !).xlim!1!1!1. 3.1 3.1. ;! ( %$ <% <%=3xy(x) = x2 + 1 :167< ;! <% <%=:3xy(x)k = xlim x = xlim x(x2 + 1) = 110 3xb = xlim (y(x) ; kx) = xlim @ x2 + 1 ; xA =3 ; x(x2 + 1)xx = 0:= xlim=lim;xx2 + 1x2 + 1% %% ' * %), * y = x: @+ )+ $ 1+ <% <%= $% $ y ; yc, y = y(x) y = x:3xy ; y = x2 + 1 ; x = ; x2 x+ 1 : x > 0 y ; y < 0 y < y, .. <% <%=1 !.
x < 0 y ; y > 0 , $), <% <%= 1 !. C, ) , jxj + + %$ <% <%= y(x) 1 '1 )(! x3 (. . 3.2). >!1!1!1!1!1!1;. 3.2168(' 2" ! "$ &'";<" " F '"" y(x) = x :x +1. y=0 { '"" " ".2||||{0 (". ( <%=+ f (x) %! % )% x0$ %), 1 '(, ! )% x0(x0) { -%( )% x0, .. (x0 ; x0 + )_(x0 ) { %+ ( %+) -%( )% x0,.. (x0 ; x0 + ) '$ )% x = x0..8M > 0 9 > 0 : 8x 2 _ (x0) jf (x)j > M <%=+ f (x) $+ & &, x ! x0:" ) 9W xlimx f (x) + (, ) <%=+ f (x) & )% x0: ?*+ $( :xlimx f (x) = 1:? .E xlimx f (x) = a 6= 0 xlimx (x) = 0 9 > 0 %, ) (x) 6= 08x 2 _(x0 ) f (x) = 1:limx x (x)!0!!0!! ! 000a!f(x)xlimx (x) = 0 = 1:169!0C( (a=0) $), ) a + 0 (=+ + (, %% $, ), 1 !.
3.2. ;!x + 1:limx 2x;23!x+1< xlim2 x ; 2 = 0 = 1: >!! 3.3. ;!x2 ; 1:limx 1 3x3 ; 5x2 + x + 1!< ?<+ # , % %% )+ $+ (0=0). ; ) , % %% $+ . @+ %+ 1,+*+ % 0 x ! 1 % ,:2;1(x ; 1)(x + 1) =x=limlimx 1 3x3 ; 5x2 + x + 1 x 1 3(x ; 1)2 x + 132!x+1= xlim1 (3x + 1)(x ; 1) = 0 = 1: >!!!(' 2" ! "$ &'" :1) xlim x;+3xx !;12+2 :2) xlim xx ;;83xx +162!2421) 12) 1:||||{170? '%) '(#, <%=! + 1( =(..8M > 0 9 > 0 : 8x 2 _(x0 ) f (x) > M <%= f (x) $ - & &, x ! x0 #xlimx f (x) = +1:.8M > 0 9 > 0 : 8x 2 _(x0) f (x) < ;M <%= f (x) $ & &, x ! x0 #xlimx f (x) = ;1: 3.4. ;!2;1x:limx 2 (x ; 2)2!0!0!< C (3!2;1x= 0 = 1limx 2 (x ; 2)2!2;1x.. (x ; 2)2 { '%) '(#+ x ! 2: %1, ) 1(+.
% %% xlim2 (x2 ; 1) = 3 '$ )%2;1x2x = 2 )( x ; 1 > 0, $) (x ; 2)2 > 0: ; jf (x)j > M f (x) > 0, f (x) > M . 3!2;1x= +0 = +1: >limx 2 (x ; 2)2? +0 ;0 + 1( 1(! =(! $( '$ )%, %! )++ (+++ 1(! =(! '%) !). 3.5. ;!2;9xlim:x 2 (x ; 2)2!!!171< 2 C ( '$ )% x = 2 )( x2 ; 9 < 0 '(x ; 9 < 0: ; jf (x)j > M f (x) < 0, f (x) < ;M ,(x ; 2)2 (, '%) '(#+ +++ =(!. "(!# ' '+ ' 1( %%! $( ;5 !2;9xlim= +0 = ;1: >x 2 (x ; 2)2! 3.6. ;!2 + 4x + 1x:limx 3 (x + 3)4 x2 + 4x + 1 ;2 !x< xlim3 (x + 3)4x = ;0 = +1: >!;!;(' 2" ! "$ &'" :1) xlim 1 ;x 3x ;2 3) xlim xx ;x2!0!0252) xlim (x ; 1);3(1x ; 2x) 4) xlim 2x(2x++x1) :2!124!;11) +12) +13) ;14) ;1:||||{172 4( + .
( ( <%=+ f (x) %! % )% x0$ %), 1 '(, ! )% x0"(a) { "-%( )% a, .. (a ; " a + "):Q a $+ <%= f (x) x0 #+_xlimx f (x) = a, 8" > 0 9 > 0 : 8x 2 (x0 ) f (x) 2 "(a):. ) " , f (x) 2 "(a) $) '$( $)! <%= f (x) % ) a:N$ < + , ) $)+<%= f (x) %( '$% % a + , $)! x 6= x0, ) '$%, % x0: 4.1. N )% x0 <%=, <% %, $'1 . , &, , #, , ) ?!0))f (x ) &'0)). 4.1< ) xlimx0 f (x) = f (x0 ) ') xlimx0 f (x) = a ) xlimx0 f (x) = a ) 9W xlimx0 f (x):C) f (x0 ) + ) xlimx0 f (x) (.
4.1 '), %%% '$( f (x) % '+ (% )%, -% )% x0 $) f (x0 ) 1 '( 1 (. 4.1 ). " ) (. 4.1 ) f (x0 ) +++ f (x) )% x0 % %% $) f (x) )% x0 +++ '$% % f (x0 ) %% ' ' _(x0):9) , ) +++ %% ). >!!!!!173M%=+ f (x) $+ )% x0 xlimx f (x) = f (x0 ):!0" 4.1 <%=+ f (x) +++ ! )% x0 ) +++ ! ($) )% x0 )+, 4.1', , . ") xlimx f (x) ,%$ <%= '$)% x0:!0(' 2" ! "$ &'"& " x = 2 F '"" f (x) lim f (x) = 4 x") f (2) = 4!) f (2) = 2) f (2) .0!2)))B , "+ " "), !), ) ? . E +; "".||||{174 .
<%=+ f (x) )% x0 xlimx f (x) = f (x0 )!0.. 1 )+( ! %! $)+ x = x0 1, $ * <%=." <%= %1 ! ! )%! ' +, .. )%, %+ %( %! %1 1 ' +.. <%=+ f (x) $ )% x0 (, ! )%), + <%=+ g(x) + )%x0 %+, ) g(x) = f (x) , % )% x0: 9) ,x g (x) = g (x0 ):xlimx f (x) = xlim!0 4.2. ;!!0x2 ; 3 :limx 1 x2 ; x ; 1!2;3x< % %% <%=+ f (x) = x2 ; x ; 1 { + )%x = 1 , (, ! )%, $) f (1) :2;31 ; 3 = 2: >x=limx 1 x2 ; x ; 1 1 ; 1 ; 1!x2 ; 2x ; 3 :limx 1 x3 ; x2 ; 2x ; 3x< M%=+ f (x) = x3 ; x )% x = ;1 % x = ;1 ( ($+, % %% )( $( '*+ 0: " 1 x + 1 +*+ % 0, % ,:2 ; 2x ; 3(x + 1)(x ; 3) = lim x ; 3 =xlim=limx 1 x3 ; xx 1 x(x ; 1)(x + 1) x 1 x(x ; 1)= ;1(;;1 1;;3 1) = ;24 = ;2:175 4.3.
;!!;!;!;!;9' , ) %*+ )+ <%=+,)*++ , ! (% )% x = ;1 $) %! ) +. > 4.4. ;!! 13;:limx 1 1 ; x 1 ; x3< 8! +++ $( , '%) '(#, <%=! ( ( 1 ; 1), $ , <%=! . @+%+ 1 % '* $ , %% * , 1 x ; 1 +*+ % 0, % ,:102;331+x+x1A@;= xlim1 (1 ; x)(1 + x + x2) =limx 1 1 ; x (1 ; x)(1 + x + x2 )!!!2+x;2(x ; 1)(x + 2) ==lim;= xlim1 (1 ;xx)(1+ x + x2) x 1 (x ; 1)(1 + x + x2)!!= xlim1 ; 1 +x x++2 x2 = ; 1 +1 +1 +2 1 = ;1: >! 4.5. ;!p1 + 2x ; 3 :limx 4x;4< Q( $( ' x = 4 '*+ ( ( ( 0=0). @+ + 1+ x ; 4 ) ($ 1 +1 1:p1 + 2x ; 3 = lim1 +p2x ; 9lim=x 4x 4 (x ; 4)( 1 + 2x + 3)x;4= xlim4 (x ; 4)(2(px1;+4)2x + 3) = xlim4 p1 + 22x + 3 = 3 +2 3 = 13 : >!!!!!176 4.6.
;!p3+8;2x:limx 0xp33+8;8xx+8;2== xlim0 p 3 2 p 3< xlim0xx x +8 +2 x +8+42x= 0: >= xlim0 p 3 2 p 3x +8 +2 x +8+4 4.7. ;!p 2 p 21+x ; 1;x :limx 0x2p 2 p 2< xlim0 1 + x x;2 1 ; x =1 + x2 ; (1 ; x2)= xlim0 2 p2 pp 2 p 22 =22x1+x + 1+x 1;x + 1;x2x2= xlim0 2 p2 pp 2 p 22 =22x1+x + 1+x 1;x + 1;x= 1 + 12 + 1 = 23 : >3!33!!3!3333!33!!3333!3333(' 2" ! "$ &'"<":1) xlim x x+ 2+x 1; 6 51x + 10 2) xlim 5xx ;;13x + 30!x+2x13) xlim 2x ; x ; 1 ; x ; 1 43!02!102p2!32!02!121) ;62) 73) 23 p4) xlim x + 13x ;;29 x + 1 p35) xlim 8 + 3xp3 ; x ; 2 :2 x14) ; 165) 0:||||{1772/ " '$)+:+(x0) { (0 x0 + ) + %( )% x0 (x0) { (0 ; x0) + %( )% x0:. 8" > 0 9 > 0 : 8x 2 +(x0) f (x) 2 "(a) ) $+ <%= f (x) )% 0 , #+:lim f (x) = f (x0 + 0) = a:x x +0;!0.
8" > 0 9 > 0 : 8x 2 (x0) f (x) 2 "(a) ) $+ <%= f (x) )% 0 , #+:lim f (x) = f (x0 ; 0) = a:x x 0;!0; <%= )% $ . xlimx f (x) $ .@! * xlimx f (x) = a (% , % * * 1 :lim f (x) = a = x limf (x):x x 0x +0!0!!00;!0 4.8. ;! f (x) )% x0 = 0 :( 2x < 0,f (x) = xa ;+x4 x > 0.< % %% > 0 <%=+ f (x) ( ; x) f (x) (a ; x) :lim f (x) = xlim0(a ; x) = a:x x0+0!! ) ) f (x) 2 + 4) = 4:limf(x)=lim(xx x 0x 0!0;!178. a = 4 * ! lim f (x) = 4:x 0.
6= 4 ! xlim0 f (x) *, %%%lim f (x) = 4 6= x limf (x) = a:x x 0x +09' , ) $) f (0) , ( %%{' '$, ) , *+ $++. >: <%= f (x) )% x0xlimx f (x) = f (x0 )($+ , 1 ( f (x0 ; 0) = f (x0 + 0) = f (x0 ):. f (x0 ; 0) = f (x0 + 0) f (x0 ), f (x0 ) , )% x0 $+)%! # <%= f (x):.f (x0 ; 0) 6= f (x0 + 0) )% x0 $+ )%! f (x):)% $ $ %) %)% <%= $+ )% I-# .. <%=+ %! % )% x0 $ %) 1 '( ! )% x0 ,+ ' $ , f (x0 ; 0) f (x0 + 0) * ' '%), )% x0 $+ )%! II-# <%= f (x): 4.9. N ( (:( 2x 0,f (x) = x4 ;+x4 x > 0.179!!!0;!!00< % %% <%= x2 + 4 4 ; x +++ , $1! )%! $ ' )% "%!%" = 0: N <%= f (x) '$ ! )%:lim f (x) = xlim0(4 ; x) = 4x 0+0!!lim f (x) = xlim0(x2 + 4) = 4f (0) = 4:N%, f (0 + 0) = f (0 ; 0) = f (0) = 4: ? (, f (x) + <%=+.
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