1. Математический анализ (850924), страница 9
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<%=+ y = e x ' x > 0. % '$, )% x = 02<%=+ y = e x %, ! 1./(! @+ %) + <%= y = f (x) ='$1 * +: 1) ( '(*+ <%= 2) ( <%= )(, )(, )( 3) +( * (%(, %,) 4) ! $+ '+ <%= 5) ! )% % <%=, %1 ( %( $)+ 6) ! % <% <%= )% ' 7) !)% )+ <% <%= ( Ox. ) % + %$ <% <%=." %) <% <%=3 ; 5x2 + 14x ; 62xy=:(15:3)4x2L ( $1! # ,.1) %(% <%=+ (15.3) + '! =('(, , $ %) )%x = 0, %! '*+ ( $(.2) M%=+ '* .3) "+ * . % %%3 ; 5x2 + 14x ; 62xlim= ;1x 04x2143;00;;;!x2 :;;0 <% <%= %( x = 0. @, $*+ f (x) = lim 2x3 ; 5x2 + 14x ; 6 =limxx x4x36;2 ; x5 + 14x2 x3 = 1 = xlim42!x = lim 2x3 ; 5x2 + 14x ; 6 ; 2x3 =limf(x);x2 x4x26;;5 + 14x x2 = ; 5= xlim44%, ) x ! ;1, x ! +1 <% <%= % y = x2 ; 54 .4) @+ ,1+ $+ '+ ) $ #! <%=3 ; 7x + 6xy = 2x3 = (x ; 1)(x2;x32)(x + 3) :% %% <%=+ $ + * x = 0, )ae * ' ,+ $% y :x(;1 ;3) -3 (-3,0) 0 (0,1) 1 (1,2) 2 (2 +1)C% +0 {+ 0 { 0+ "$:'"$:'"$<%= ccc5) N$ ! '= % (, ) <%=+ * )% %:49 % x = ;3, ) f (;3) = ; 12% x = 1, ) f (1) = 54 x = 2, ) f (2) = 98 .!1!1!1!1!1!10001446) @+ ,1 + ,+ + % ) $ 7(x ; 97 )7x;9y = x4 = x4 :00N+ , ) <%=+ $ * )% x = 0, ) * ,+ $% y :x(;1 0) 0 (0 9=7)9/7(9=7 +1)C% {{0+"%("%("%( ' " ( (N$ !) , ) <% 9 '=!!! 913<%= 99 ' )% 7 f 7 .
9, ) f 7 = 756 .00007) 9+ ! )% )+ <% ( Ox. 6)% * %+ +2x3 ; 5x2 + 14x ; 6 = 0:A % (, )2x3 ; 5x2 + 14x ; 6 = 2(x ; 21 )(x2 ; 2x + 6):%(% % ! ,) (x2 ; 2x + 6) *, %! ( % =( )), *! %( x = 12 , %1 !) <% <%= % ( Ox )% 2 0 : ) %$ ! <%= (. 15.4).145;(;3) { 1 { 2 { 9=7 { . 15.4 1. 0 "+ " " (a b)(c . 15.1).2.
>"" " "" '!" '"" ? (c . 15.3 15.4).3. 0 "+ !; '!" (c . 15.5).4. 0 "+ " '!" (c . 15.3).5. " " " " x ; 3x + q , ; " x ; 3x + q = 0 ; .335. 6" ( y = x ; 3x + q " F " F . ( y = 3x ;3 " 0 (!; F ", " 14.4):3x ; 3 = 0 x = 1 x = ;1 x = 1: " F " ( . 14.5), " F " "? . @ F " !" ;" "" y = 3(x ; 1)(x + 1) "".3022201;23" !" , x = ;1 " " " "+" " "{",", " x = ;1 { " " " y(x).
x = +1" " " "{" " "+", ", " x = +1 { " " y(x).11146116" " y = x ; 3x + q F; ";.3y(;1) = 2 + q y(+1) = q ; 2:3" !" , + (1 ":1) q < ;2:3" " y(;1) < 0 y(+1) < 0, '" ; " ' (1 !" :;;;; " x ; 3x + q = 0 .2) q = ;2:3" " y(;1) = 0 y(+1) < 0, '" ; " ' (1 !" :3" x ; 3x + q = 0 " .3) ;2 < q < 2:3" " y(;1) > 0 y(+1) < 0, '" ; " ' (1 !" :3 " x ; 3x + q = 0 .4) q = 2:3" " y(;1) > 0 y(+1) = 0, '" ; " ' (1 !" :3 " x ; 3x + q = 0 " .5) q > 2.3" " y(;1) > 0 y(+1) > 0, '" ; " ' (1 !" :3147; " x ; 3x + q = 0 .3" !" , " ; ;2 < q < 2.3148 1( ( "Co%( $)! <%= xn = f (n) ( n $+ (.Q $+ ( 1 x2 : : : xn : : :, + ' " > 0 * % ) N , ) jxn ; aj < " n > N .
" ) #nlim xn!1= a:N($+ )% % '* 8 ()+ " + ' " " + ,") *+ 9 ()+ "*" "! +"), o ( 1 $(%%nlim xn = a 8" > 0 9N : 8n > N jxn ; aj < ". 1.1. @%$(, )1 = 0:limnn1< n ; 0 = n1 < " n > 1" . , 1 N = 1" )118" > 0 9N = " : 8n > N n ; 0 < ":6 $), )1 = 0: >limnn 1.2.
@%$(, )1 = 0:limnn21< n2 ; 0 = n12 < ", n2 > 1" n > p1" . 1 N = p1" 149!1!1!1!1)118" > 0 9N = p" : 8n > N n2 ; 0 < " =) nlim0 n12 = 0:(C( =) $), ) $ .) >. xn = C (n = 1 2 : : :), (( $+ . ) (! 1 ($( * .. 9 nlim xn nlim yn C { ++, 1) nlim C = C 2) nlim C xn = C nlim xn3) nlim (xn yn) = nlim xn nlim yn 4) nlim (xn yn) = nlim xn nlim ynlim x5) nlim xy n = nlim yn ec yn 6= 0 nlim yn 6= 0:nnn!!1!1!1!1!1!1!1!1!1!1!1!1!1!1!1 1.3.
;!2+n;1nnlim 2n2 ; n + 1 :< Q( $( +++ ) ! . @+ , n2 ( # () )+, )1 = 0 lim 1 = 0lim(: : 1:1 1:2)nnnn2)2+n;12lim (1 + 1=n ; 1=n2)n1+1=n;1=nnnlim 2n2 ; n + 1 = nlim 2 ; 1=n + 1=n2 = lim (2 ; 1=n + 1=n2 ) =n!1!1!1!1!1!1!11 + nlim 1=n ; nlim 1=n2 1 + 0 ; 0 1= lim 2 ; lim 1=n + lim 1=n2 = 2 ; 0 + 0 = 2 : >nnnnlim!1!1!1!1!1!11500 " (( fxng $+ & , nlim xn = 0: 1.4. @%$(, ) ((xn = 2n2n;;n1+ 1 (n = 1 2 : : :) +++ '%) !.< %1, )n ; 1 = 0:limx=limnnn2n2 ; n + 1"1 + xn + '! # , ).
?#+ ( , ), n2. 8$ )( $( n2 ($+ < # + ) 2n;11=n;1=n0 = 0:lim=lim=n2n2 ; n + 1 n 2 ; 1=n + 1=n2 2? (, (( xn { '%) +. > (( fxng $+ # , 9M > 0 : 8n jxnj M: ), (! +++ (fsin ng(8n j sin nj 1)farctg(1=n)g (8n 0 < arctg(1=n) =4 =) j arctg(1=n)j =4)farctg ng(8n j arctg nj =2):!# # ( ( +++*!+ )!) 1 1( ((xn = n (n = 1 2 : : :):D% ' '(# ' ) M ,xn > M n > M:151!1!1!1!1!1? (, * % ) M > 0, )' jxnj M +( 8n.
% %, ' ($( * .. fxng { '%) + ((, fyng { )+, (( fxnyng +++ '%) !. 1.5. ;!n;1nlim 2n2 ; n + 1 sin n:< % %% (xn = 2n2n;;n1+ 1 yn = sin n (n = 1 2 : : :)+++, c, '%) ! (. 1.4) )!, (( fxnyng +++ '%) ! , (,n ; 1 sin n = 0: >limn2n2 ; n + 1!1!10 (". " (( fxng $+ & &,, 8M > 0 9N : 8n > N jxnj > M:B ) #nlim xn = 1 +, ) ( '%)!. 1.6. @%$(, )2nlim n = 1:pp< jn2j = n2 > M , n > M . 1 N = M )p8M > 0 9N = M : 8n > N jn2j > M =) nlim n2 = 1: >H %$(, ) 8 > 0 a 6= 0nlim an = 1:!1!1!1!1152@+ ++ '%), ' ($( :. 9 nlim xn 6= 0, !1nlim yn!1= 0 yn 6= 0xnnlim yn!1 1.7. ;!= 1:2 ; 2n + 5nnlimn+4 :< @+ ,1 + ($+ <!# !:2 ; 2n + 52n1;2=n+5=nnlimn + 4 = nlim 1=n + 4=n2 = 1% %%!25nlim 1 ; n + n2 = 1 6= 01 4 !nlim n + n2 = 0: >!1!1!1!1!1 ( fxng fyng $+ 0, xn = 1:limnyn" ) #xn yn: 1.8.
@%$(, )!1(n3 + n ; 1) n3 (8n3 ; n2 + 4) 8n3:< @!(, $ ) $ '! n3, )2 ; 1=n33+n;11+1=nn= 1nlimn3 = nlim1!1!11533 ; n2 + 438n8;1=n+4=n= nlim= 1:nlim8n388 , = $) %(*, (!. >" '* ) 1 > 2 > : : : > m a1 6= 0!1!1(a1n + a2n + : : : + amnm ) a1n :121(1:1) ) (! )( ' ($( * .. xn xn yn yn yn 6= 0 9 nlim xy n , 000!1n0xnxnnlim yn = nlim y :n0!1!1(1:2)0 1.9. ;!3 ; n2 + 48nnlim n3 + n ; 1 :33 ; n2 + 48n8n< nlim n3 + n ; 1 = nlim n3 = nlim 8 = 8: > 1.10. ;!2;3nnlim n4 + n2 + 1 :!1!1!1!1!1< :)+ (1.1) $++ )( $( % (, )22;3n1 = 0: >n=lim=limlimnn4 + n2 + 1 n n4 n n2 1.11.
;!4 ; n2 + 4nnlim 9n2 ; 2 :!1!1!1!1424< nlim n 9;n2n; +2 4 = nlim 9nn2 = nlim 19 n2 = 1: >!1!1!1154? 1.9{1.11 $+ +(, ) Pm (n),Pk (n) { ) ! m k coo, 8> 0 m < kPm (n) <a m = k(1:3)nlim Pk (n) = >: 1 m > k.!1C ( { # %<<= #, +, ).
1.12. ;!2 ; (n ; 1)2(n+1):nlim2n + 5< ; (, ) )( +++ ) ! . " !1(n + 1)2 ; (n ; 1)2 = (n2 + 2n + 1) ; (n2 ; 2n + 1) = 4n:% '$, )( $( +++ ) ! # %<<= #, +,(n + 1)2 ; (n ; 1)2 = 4 = 2limn2n + 52 $ % %1 $ ,!1(n + 1)2 n2 (n ; 1)2 n2 ' % ( $(, % %%(n + 1)2 ; (n ; 1)2 6 n2 ; n2 = 0: >C, ) xn n yn n (c1xn + c2yn) (c1 + c2)n(1:4)#( (c1 + c2) 6= 0:(. (1 + 2) = 0, # (c1xn + c2yn)=(c1 + c2)n + .)155(' 2" ! "$ &'" (1 :5n + 2 (n + 2) ; (n ; 1) 1) nlim 43nn +4)limn+ 3n ; 43n3n ; 2 (n + 3) ; (n ; 3) 2) nlim n(n; ;5)limn2)(n + 3) + (n ; 3); (3n + 1) :6) nlim (n + 1)3) nlim n 2+n2n; n;;n 2; 1 3n + n222!12!132!123!11) 43 2!133223223!13) 12) 024) 06) 13 :5) 9||||{ *1 (, 1!*"8 )+ #+ , ) ++ '!, 1*, =(.
1.13. Ha!p 38n ; 2n + 3 :limnn;3< 8$ )( $( ' n. :)+,vvp 3)u3 ; 2n + 3uu8n ; 2n + 3 = u8ntt8 ; 2 + 3 =nn3n2 n3)qp 3p2 + 3=n38;2=n8n;2n+38 = 2: >=lim=limnnn;31 ; 3=n1 1.14. @%$(, )pp 49n + n + 4 3n2 8n6 ; n4 + 2 2n2:3!1333333!1!13p9n4 + n + 43n2vuu 41= nlim t 9n + 4n + 4 =< nlim3nvuu= 31 nlim t9 + n13 + n44 = 13 3 = 1:156!1!1!1pvu6 ; n4 + 2 16 ; n4 + 2u8n8nt= 2 nlim=26nlim2nnvuu1= 2 nlim t8 ; n12 + n26 = 12 2 = 1:N$ = , #! %$.
>C, ) '*, xn n, 33!1!13!1pm x n=m:n 1.15. ;!(1:5)p 49n + n + 4 :plimn8n6 ; n4 + 23!1< ++ $ (! % ) (< 1.2) )+ %( (!, , 1.14, )p 429n+n+43n3: >p=limlim=n8n6 ; n4 + 2 n 2n2 2 1.16. ;!p2n + 7nnlim 2n + 1 :< :)+ #+ (1.5) (1.2), )!13!13!1p2=32 + 7nnn1nlim 2n + 1 = nlim 2n = nlim 2n1=3 = 0: > 1.17. ;!p7p2n+2;n + 1:pplimnn3 + 1 ; n4 + 2pp< % %% n7 + 2 n7=4 n2 + 1 n2=33!1!1!1435!143p5n4 + 2 n4=5p157n3 + 1 n3=2a (1.1),7 > 2 =) (pn7 + 2 ; pn2 + 1) n7=44 33 > 4 =) (pn3 + 1 ; pn4 + 2) n3=22 5p7p27=4n+2;n+1n1=4nlim pn3 + 1 ; pn4 + 2 = nlim n3=2 = nlim n = 1: >435345!1!1!1C 1.15{1.17 %$ , ) (1.3)''*+ '!, 1*, =(, m u ( %$ ! nm u n %,) $ (+,, { #%<<= , ()*,+ $) %! $ %<<= #, +, %, 1+,). 1.18.
;!pnlimpn2 + 3n + 1 ;!1pn2 ; 5:p2n ; 5 n pp2 2n + 3n + 1 ; n ; 5 6 n ; n = 0< C ( n2 + 3n + 1 n(c. 1.12 (1.4)), %+ $ % ' % #') $(.@+ )+ ) $( %! 1 $ +1 1 ( %!). :)+, )pp( n2 + 3n + 1 + n2 ; 5) (n + n) = 2n)pp22nlim n + 3n + 1 ; n ; 5 =p2p2p 2p2(n+3n+1;n;5)(n+3n+1+n ; 5) =p2p2= nlimn + 3n + 1 + n ; 5158!1!12 + 3n + 1) ; (n2 ; 5)(n3n + 6 = lim 3n = 3 := nlim=limnn2n2n2n 2( $ %! % , 1 )+ ' .) > 1.19.
;!p2p2lim(n+1;n ; 1):n< ++ , )! ($ 1.18, )p2p2lim ( n + 1 ; n ; 1) =p 2np2p 2p2(n+1;n;1)(n+1+n ; 1) =p2p2= nlimn +1+ n ;12 + 1 ; n2 + 1np 2 = nlim p 2 2 p 2 == nlim p 2n +1+ n ;1n +1+ n ;1= nlim 22n = 0: > 1.20. ;!2 (pn3 + 4 ; n):limnnp3< C ( n + 4 n, , 1 % $+( ($+, % %%p( n3 + 4 ; n) 6 (n ; n) = 0(c. *! ). " ) 1 ! % p3pn + 4 + n n3 + 4 + n2) $( %' )p( n3 + 4)3 ; n3 = n3 + 4 ; n3 = 4 p24n23qpn + 4 ; n = nlim=nlim n( (n3 + 4)2 + n n3 + 4 + n2)24n= nlim n2 + n2 + n2 = 43 : >!1!1!1!1!1!1!1!1!13!1333333!1!1!115933(' 2" ! "$ &'"1)2)3)4)5) :6n + 1 plim4nn + 3n ; 1p7 n + 1lim p3n3n ; 1pn+n+1 lim p3nn ;n +2p qp3 p3lim n + 2n + 1 ; n + 2 n(n + 1) ; n nplimn+4n+2;nn4!12!15!13332!12!1p6) nlim n!1ppn+2; n;3 :7) nlim qp3(n + 2) ; 3 n ; 3!1221) 6 2) 0 3) 0 4) 0 5) 2 6) 1 7) 0:||||{160 2( + 9 (( <%=+ f (x) x > x0.
Q a $+ <%= f (x) +1, + ' " > 0 *% ) N , ) jf (x) ; aj < " x > N , ..lim f (x) = ax + *( )%, %:lim f (x) = a 8" > 0 9N : 8x > N jf (x) ; aj < ":x +!!11 2.1. @%$(, )1 = 0:limx + x1< x > 0 x ; 0 = x1 < ", x > 1" .N = 1" , )!1, 1118" > 0 9N = " : 8x > N x ; 0 < ":1 = 0: >6 $), ) x lim+ x9) , + <%= f (x) +1 ''* + ( ff (n)g )!, % + $)+ <%= 8x > x0, (% (,x = n.
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