1. Математический анализ (850924), страница 2
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1.2). " " % fxng nlim!1 xn=1 , # . , 1 " " %, " #. ,1 , .= , " , * , 1.5 + * " . 1.12.4+14nn3nlim!1 n + 5 = nlim!1 n = nlim!1 n = 1:+ . 8 " " %, . 6 , " 19 " " %. 6, n : : :fn sin ng10;2030;4:::nsin22 ( jxnj M M " % n), " " % ( jxnj > M n). 1.8.
fxng - ( #), 8n xn xn+1 (xn xn+1).. n (" ).8 8n xn < xn+1 (xn > xn+1) fxng ( #). 1.9. $ 1 "1 , 1 "1, # . 1.6., # .? .!n )(14 1 + n . *, 1 . & " 6 , !n1n ; 2) 1 +xn = 1 + n = 1 + n n1 + n(n2!; 1) n12 + n(n ; 1)(3!n3!n(n;1)(n;2)(n;(n;1))111++ nn = 1 + 1 + 2! 1 ; n +n!20!!! n ; 1!11211+ 3! 1 ; n 1 ; n + : : : + n! 1 ; n 1 ; n :!k n n + 1 * " 1 ; n ( 1 ; nk < 1 ; n +k 1 ) " (* ). +8n xn < xn+1 ) fxn g { 1 .?, , 1 ; nk < 1 (k = 1 2 : : : n ; 1) 1=11 (n = 3 4 )<n! 1 2 3 n 2n;1 1 1!1111xn < 1 + 1 + 2! + 3! + + n! < 1 + 1 + 2 + 22 + + 2n;1 = 1 !n1; 21 = 3:= 1+<1+11 ; 122( & .)2 ",0 < xn < 3:= , fxng { .
1.6. . ( % e!n1nlim!1 1 + n = e:'* , e { 10;15e = 2 718281828459045:21$-+ ,&+1. .2. a fxng, jxn ; aj < " n > N ? (. 1.2).3. ! " ? (. #1.1 1.1).4. % % # ? (. 1.1).5. ! (. 1.3).6. )*%, n sin n2 ! (. 1.6). + ** n " jxnj 100.7. % ! ? (. # 1.2).8. "* (. 1.4).9. *% "* # ! "* 0 (.
1.3).10. 1* #0 2*? (. 1.6).11. + ** # C " (2n2 + 3) Cn2? (. 1.7).12. 1*0 % # 8n2 ; n + 4 ?limn!1 2n2 + n ; 4(. 1.5). 4 ?13. "* "5 (. 1.7).14. % "* "5 " !? (. 1.3, 1.7 1.10).15. 1* #0 ? (. 1.8,1.9). 1 n16. 4 nlim!1 1 + n ? 4 # " 6 2! ? (. 1.6).2. , jxn ; aj < " n > N .7 ( *! N 6) ".6. + n = 2k + 1 k 50 (k { :).11. + C = 2.12. + 4.22 2 456 7 @ * x, 1: a < x < b (a b)J a x b La b]J a x < b La b)J a < x b (a b]J x > a, x < a, x a, x a # (a +1) (;1 a) # La +1) (;1 a]:'* " # (;1 +1).$ , " -.
"* , " , *1 *1 a b , " ( (;1 +1)) { ,*1 *1 a. ) (;1 +1) " .A" (a b), *1 x0 (x0 2 (a b)) " x0 " O(x0). O(x0) x0 " x0 " O_ (x0 ).) (x0 ; x0 + ) " - x0 " (x0).- x0 x0 " - x0 " _ (x0 ). 2.1. & f (x) (x0 +1).D a & f (x) x ! +1 ( +1 ), 8 " > 0 9 N : 8 x > N jf (x) ; aj < ":23 ":a = x!limf (x):+1$ . ? + , y = a + " y = a ; ", " "- ( .
2.1).;. 2.1E 8x > N jf (x) ; aj < " , x > N & & f (x) "- . 2 ", a = x!limf (x), x +1, .. +1 x, ( x N ), & & f (x) "- .2 " * " (* ), & f (x) " a " % x. $ , " a x * x {" *. 2.1. 4 &f (x) = x1 : x ! +1 * . *, 1 = 0:limx!+1 x " " > 0.1 x > 0 x ; 0 = x1 < ", x > 1" .11* N = " . 2 8x > N x ; 0 < ":242 ", 111 = 0:8" > 0 9N = " : 8x > N x ; 0 < " ) x!lim+1 xO& & x1 x > N "- , y = " y = ;" ( . 2.2).;.
2.2 x ! +1 & 1 , % " > 0, x " %, N . $ "1 , + , N " % " N . 2.2. P f (x) * E , 9M > 0 : 8x 2 E jf (x)j M: 2.2. P sin x (;1 +1), 8x 2(;1 +1) j sin xj 1 (M = 1): 2.3. P f (x) = x1 (1 +1), 8x 2 (1 +1) x1 < 1 (M = 1):6 + & ( ) (0 1), 1 M > 0, " 8x 2 (0 1) x1 M: 2.3 , * & * " * " , { .25 2.3.P f (x) x ! +1 , 1 N , & f (x) (N +1). 2.4.
P x1 x ! +1, ( . 2.3) (1 +1) (N = 1): 2.1. 9 x!lim+1 f (x) = a, ./ f (x) x ! +1.. ? " = 1 9N : 8x > N :: 8x 2 (N +1)jf (x) ; aj < 1: jf (x)j = j(f (x) ; a) + aj jf (x) ; aj + jaj < 1 + jaj:* M = 1 + jaj, 8x 2 (N +1) jf (x)j < M.. & f (x) (N +1) , , x ! +1. 2.1 & +1 1.2 1 % , jf (x) ; aj < " " f (x) a " # x > N , # n > N . ? + " , . A , & +1. ? % .
2.4. & f (x) (;1 x0):D a & f (x) x ! ;1 ( ;1 ), 8" > 0 9N : 8x < N jf (x) ; aj < ": ":a = x!;1lim f (x):26 2.5. ?*, 1 x < 0 x ; 0 = ;1x= ,1 = 0:limx!;1 x< ", ;x > 1" x < ; 1" .11lim x1 = 0:8" > 0 9N = ; " : 8x < N x ; 0 < " ) x!;1O& & x1 x < N "- , y = " y = ;" ( . 2.3).;. 2.3$ 2.4 8" > 0 8x < N jf (x) ; aj < ", x ! ;1, .. % x, f (x) " a. , " " " > 0 f (x) " a % ", x % N .
2.5. 8 1 & f (x) x ! +1 x ! ;1 * a, a & f (x) x ! 1 ( 1 ). ":a = xlim!1 f (x):1 = 0, lim 1 = lim 1 = 0 ( . 2.6. xlim!1 xx!+1 x x!;1 x2.1 2.5).O& & x1 "- , y = " y = ;" x > N = 1" x < ;N = ; 1" , .. jxj > N( . 2.4).27;. 2.4<, "1, a = xlim!1 f (x), 8" > 0 9N > 0 : 8jxj > N jf (x) ; aj < ":$ , 8" > 0lim f (x) = a ) 9N1 : 8x > N1 jf (x) ; aj < "Jx!+1(2:1)lim f (x) = a ) 9N2 : 8x < N2 jf (x) ; aj < ": , jf (x) ; aj < " 8x > jN1j 8x < ;jN2j, N = max(jN1j jN2j), 8x > N 8x < ;N , ..
8jxj > N . 2", (2.1) . ", (2.1) , x!;1jf (x) ; aj < " 8x > N 8x < ;N:= " , x!limf (x) = a x!;1lim f (x) = a, +1xlim!1 f (x) = a:2 ", (2.1) (2.5).<, , x!limf (x) 6= x!;1lim f (x), xlim!1 f (x) 1 +1 ( " 1 , +1 ;1 " ").2 & ;1 1. 2.7.2x1 = 1 = 1:lim=limx!1 x2 + 1 x!11 + x12 1 + 028 2.8.1xx = 0 = 0:lim=limx!1 x2 + 1 x!11 + x12 1 + 0 & " &, " * (a +1), (;1 a) (;1 +1).
? & & * * . & f (x) (a +1). 4 & y = kx + b:(2:2)$ "(x) = f (x) ; (kx + b):(2:3)$ * & & * x ( . 2.5).;. 2.5 2.6. 8 x!lim+1 (x) = 0, (2.2) & & y = f (x).8 (2.2) & &, x ! +1 & "* .> & *, "1 , , & * ( * , & : f (x) = kx + b).29 (2.2) & y = f (x).2 (2.3) f (x) = kx + b + (x) x!lim(x) = 0:+14 1 :!11f(x)= x!limk + b x + x (x) = k + b 0 + 0 0 = kJlim+1x!+1 xlim (f (x) ; kx) = x!lim(b + (x)) = b + 0 = b:x!+1+12 ", (2.2) &y = f (x), 1 , 1 :f (x) b = lim (f (x) ; kx)):(2:4)k = x!limx!+1+1 x= ": 1 (2.4), (2.2) & y = f (x), (2.4) lim (x) = x!lim((f (x) ; kx) ; b) = b ; b = 0:+1x!+18 " (2.4) 1 , " 1, & y = f (x) ( " 1 ").
2.7. 8 x!;1lim (x) = 0, (2.2) & y = f (x).8 xlim!1 (x) = 0, (2.2) ( ) & & y = f (x). k 6= 0 #, k = 0 { #.> %* , 1 (2.4), x ! +1 x ! ;1 x ! 1 " , " (2.2) " & y = f (x).30 2.9. 4 &3xf (x) = x2 + 1 :6 &. ) 2.7 2.8, :23xxk = xlim!1 x2 + 1 = 1J!1 x(x2 + 1) = xlim10 3xA = lim ;x = 0:@;xb = xlimx!1 x2 + 1!1 x2 + 1= , y=x ( ) & &.<, jxj x2 " 13xf (x) 1 = x3:( , " x = 0 & & " " ". + , * + & & ( . 2.6).;. 2.6$-+ ,&+1.
*: f (x) x ! +1 x ! ;1x ! 1 (. 2.1, 2.4, 2.5 (2.1)).2. +8 0 2m x < 2m + 1<f (x) = : 1 2m + 1 x < 2m + 2m = 0 1 2 : : :31 x!limf (x) = 1?+13. ! *: f (x) E, x ! +1, x ! ;1, x ! 1 (. 2.2, 2.3 3).4. *: f (x), 2, ! " E, x ! +1, x ! ;1, x ! 1?5. !* *: , , , *, !# (. 2.6 2.7).2. ; ", * ** " = 1 jf (x) ; 1j < " " 2m x < 2m + 1 (m = 0 1 2 : : :) , , " = 1 6 * N ," jf (x) ; 1j < " x > N .+ ! ** ! *:f (x) x ! +1.3. <*: f (x) # ! x ! ;1 ( x ! 1), 6 * N > 0, *: f (x) ! (;1 ;N )( (;1 ;N ) (N +1)), ..8x < ;N (8 jxj > N ) jf (x)j M:4.
<*: f (x) " ! E x ! +1, * ** 8x 2 E 8x 2 (0 +1) jf (x)j 1 (M = 1).=! x ! ;1 x ! 1 , * ** *:f (x) x < 0.32 38 456 @ , x x0 x ! x0 ; 0, x , %, x0. 3.1. & f (x) (x1 x0).D a ./ f (x) x ! x0 ; 0 ( x0 ), 8" > 0 9N : 8x 2 (N x0) jf (x) ; aj < ": ":a = x!limf (x) a = f (x0 ; 0):x ;008 a = x!limf (x), x ! x0 ; 0 & "x ;0 a , % " > 0, x (N x0) ( . 3.1).0;. 3.16 , & " (x1 x0) * * N = x1.
6 "1 N " % " N . + (N x0) * x 2 (N x0) " x0.( , , & f (x) " a x < x0, " x0. $ , " a ".= 3.1 2.1 & x ! +1. 8 , x > N () x 2 (N +1)33(.. + ), 2.1 * :a = x!limf (x) 8" > 0 9N : 8x 2 (N +1) jf (x) ; aj < ":+1$ & 3.1. @ + & f (x) x ! x0 ; 0. + " &, 1 , & x ! x0 ; 0, (b x0).@ , x x0 x ! x0 + 0, x % , " %, x0. 3.2. & f (x) (x0 x1).D a ./ f (x) x ! x0 + 0 ( x0 ), 8" > 0 9N : 8x 2 (x0 N ) jf (x) ; aj < ": ":a = x!limf (x)x +00a = f (x0 + 0):8 a = x!limf (x), x ! x0 + 0 & f (x) "x +0 a , % " > 0, x (x0 N ) ( .
3.2), , & " a ( , *" , -" a) x > x0, " x0.0;. 3.234A , x < N () x 2 (;1 N ) 3.2 & f (x) x ! ;1.2 & f (x) x ! x0 + 0 ( + 1" &f (x) x ! x0 + 0). & f (x) x ! x0 ; 0 x ! x0 + 0 + &. * a,8 " lim f (x) = x!limf (x) = ax +0x!x0;00(3:1) a ( )& f (x) x0. , " & , &, ( . * 3.3).) (3.1) , 8" > 0 9N1 N2 : 8x 2 (N1 x0) 8x 2 (x0 N2) jf (x) ; aj < ":( . 3.3).;.
3.3* = min(x0 ; N1 N2 ; x0) ( = x0 ; N1). 2, x 2 _ (x0), x 2 (N1 x0) x 2 (x0 N2). = ,8x 2 _ (x0) jf (x) ; aj < ":352 ", (3.1) , 8" > 0 9 _(x0 ) : 8x 2 _(x0) jf (x) ; aj < "(3:2) ", (3.2) (3.1).$ , " > 0, 1 (3.2) _(x0) * N = x0 ; . 2, x 2 (N x0), x 2 _ (x0) , (3.2),jf (x) ; aj < ".2 ",8" > 0 9N = x0 ; : 8x 2 (N x0) jf (x) ; aj < " ) x!limf (x) = a:x ;00>, N = x0 + , * , lim f (x) = a:x!x 0+), (3.2) (3.1). ) " &. 3.3.
& f (x) O_ (x0).D a () ./ f (x) x ! x0 ( x0), 8" > 0 9 _(x0 ) : 8x 2 _(x0 ) jf (x) ; aj < ": ":a = xlim!x f (x):4 (3.2) (3.1) * " & 0 3.1. # xlim!x0 f (x), # a, , # # # a lim f (x) lim f (x).x!x0;0x!x0+0? % *. 3.1. E " jf (x) ; aj < " " f (x) 2 (a ; " a + ") f (x) 2 "(a) ("(a){ "- a). 3.2. & f (x) x0 f (x) x0 (x ! x0, x 6= x0), 36* " * .
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