1. Математический анализ (850924), страница 4
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5.1 1 2 (0 1), f ( ) = 0, .. (0 1).6 5.1 " % "* % , " ( ). , % , 5.1. @ , + (0 1). 4 + x = 21 1!1!5& f (x): f 2 = ; 8 < 0: 2 f 2 < 0 f (1) > 0 1 ! 2 1 .1 !2 2 1 . $ 3!f 4 < 0, f (1) > 0.
< 573 !7!> 0,4 1 !: 1 f 8 > 0 f (1)3 7!3 f 4 < 0, + , 4 8 :* + , , * , *1 , . 8 , 2 ( ), , * +2 , ** , % % ;2 ( . 5.4).;. 5.4 5.2 ( - # # ./). ./ f (x) 2 C La b] # - f (a) f (b).(a b) --( 1 5.1. 6 5.1 , +.= 1. 8 , ,f (a) < A < f (b) A { * * f (a) f (b). $ " A 9 c 2 (a b) : f (c) = A .. & & y = f (x) (a b) " , A ( .
5.5).;. 5.558 5.3.D M " % & f (x) La b], 8x 2 La b] f (x) M 9 2 La b] : f ( ) = M: ": 5.4.M = xmax2ab] f (x):D m % & f (x) La b], 8x 2 La b] f (x) m 9 2 La b] : f ( ) = m: ":m = xmin2ab] f (x): <=$&& 5.3. ./ f (x) 2 C La b], La b] * * .? . 5.2. E* * " & , ! . 5.2. P tg x ; 2 2 , " %, % ( . 5.6).;. 5.6 5.3. P f (x) = x (0 1] , (8x 2 (0 1] jf (x)j 1), max f (x)x2(01]59 min f (x)x2(01];( f (x) > 0 , 1 (0 1], f (x) " " 0) ( .
5.7).. 5.7$-+ ,&+1. # *: * *: 2* ? 7 "?2. *:, #* (c. 5.2).3. + *:, #* (c. 5.1{5.3).4. "5! 5! # *: #* (c. 5.2, 5.3).5. ?#% ! 5 (c. 5.1 5.1).1. (c. 5.1 (14) *: @4). =" *% (o" . %).60 67 8 456 6.2.P (x), O_ (x0 ), " & x ! x0, xlim!x0 (x)= 0:6 "" ; " + :8" > 0 9_ (x0) : 8x 2 _ (x0) j(x)j < ": 6.1. P (x) = x { " x ! 0, lim x = 0:P sin x { " x ! 0, x!0lim sin x = sin 0 = 0:x!0P cos x { " x ! 2 , = 0:limcosx=cosx!2P cos x " x ! 0, 2lim cos x = cos 0 = 1:x!0 6.1.
* & * " " , * " (" ).@ & " 1 : x ! x0 (x) (x) { # ./, f (x) { x ! x0 , ./(x) + (x) (x) ; (x) (x) f (x) (x) f (x) # x ! x0 :61? + 1 ( . 1.3). =& & % *, " . 3.2 3.3 + & * . 1 , " & x ! x0 x ! 0 " & x ! x0. ( 1.3).2 & f (x) x ! x0, 9M > 0 _1(x0) : 8x 2 _1(x0) jf (x)j M:< " > 0. 2 & (x) { " x ! x0, M"9 _2(x0) : 8x 2 _2(x0) j(x)j < M" :* = min(1 2). 2 8x 2 _(x0) " jf (x)j M j(x)j < M" , ,8x 2 _(x0 ) j(x)f (x)j < M" M = ":2 ",8" > 0 9_ (x0) : 8x 2 _(x0 ) j(x)f (x)j < " + , (x)f (x) { " & x ! x0: 6.2.1 = 0limxsinx!0x1 x ! 0 & x { " , sin x 1 8 x 6= 0, , & sin x1 x ! 0, ,x sin x1 { " & x ! 0:62A" , + 1 , ,1limsinx!0x 1 . , x ! 0 x1 ! 1 sin x1 " * ;1 +1.
+, , " = 21 1" sin x ; a < 12 a " 8x 2 _(0) " _(0) .) 1.4 . 6.1. , # xlim!x0f (x) = a , # ./ f (x) - # f (x) = a + (x) (x) { ./ x ! x0 :( " , , &, & " .$" 6.2. ./ f (x) * x ! x0,1 ( # f (x) ./ f (x)) { x ! x0 :. < " > 0. 2 & f (x) { " " % x ! x0, M = 1" : ./9_ (x0 ) : 8x 2 _ (x0) jf (x)j > 1" :63 , 8x 2 _(x0) -, f (x) 6= 0 f (1x) 1 , -, f (x) < ":2 ",18" 9_ (x0) : 8x 2 _(x0) f (x) < "..
f (1x) { " & x ! x0: 6.3. ./ (x) { x ! x0 8x 2 _1(x0) (x) 6= 0 ./ (1x) * x ! x0 :. < M > 0. 2 (x) { " & x ! x0, " = M1 :9_2(x0) : 8x 2 _2(x0) j(x)j < M1 :* = min(1 2). 2 8x 2 _ (x0) -, (x) 6= 0 (1x) 1 , -, (x) > M:2 ",18M > 0 9_ (x0) : 8x 2 _(x0) (x) > M.. (1) { " " % & x ! x0. " " % & ( " ) " 1 . 6.4.
xlim!x0 f (x)= a 6= 0 a xlim!x (x) = 0 8x 2 _(x0) (x) 6= 0064f (x) = 1:limx!x (x)0.(x) = 0 = 0limx!x f (x)a0(x) { " & x ! x . 2 0f (x)8x 2 _(x0 ) (x) 6= 0 8x 2 _(x0) f ((xx)) 6= 0: 6.3 & f ((xx)) " " % x ! x0, . 6.3. 6x2 + 4 :limx!2 x ; 22 ..2 + 4) = 8 6= 0 lim(x ; 2) = 0 x ; 2 6= 0 x 6= 2lim(xx!2x!2x2 + 4 = 1:limx!2 x ; 2$ (x) (x) { " & x ! x0: 6.2.
8 (x) = 0limx!x (x) , (x) { " % (x), (x) { " % (x). 6.4. (x) = x2 (x) = x { " & x ! 0: 2 2xlim = xlim!0 x = 0 x!0 x650 x2 { " % x, x { " % x2.($"1 " x ! 0 & xn xm % " , " %.) 6.2. 8 (x) = 1limx!x (x) (x) = 0limx!x (x)( . 6.2) , , (x) { " * (x), (x) { " #* (x).
6.3. 8 (x)xlim!x (x) = c 6= 0 1 , (x) (x) { " . 6.5. (x) = sin x (x) = x { " & x ! 0. 2 sin x = 1limx!0 x( . (3.5)), sin x x { " . 6.3. 8 (x)limx!x (x) 1 , " & (x) (x) #. 6.6. (x) = x sin x1 ( . 6.2) (x) = x { " & x ! 0: 2 x sin x11 9F=limsinlimx!0x!0xx000066 " x sin x1 x .$ % o("" ).8 (x) = 0lim(6:1)x!x (x) , " " % = o( ) x ! x0:< & (x) (x) " " .8 * " x ! x0, = o( ) ()() (x) { " % (x): 6.7. 2 x2 = 0 lim x3 = 0 : : : lim xn = 0 : : :limx!0 xx!0 xx! 0 x x2 = o(x)J x3 = o(x)J : : : J xn = o(x)J : : : x ! 0:) + " ", , , x2 = x3 = ::: = xn = :::( : ). ? , e o(x) &, &.$ "1 o( ) x ! x0 * & (x), (6.1), * & (x) + * = o( ) * 2 o( ).= 1 *:1: 8 1 2 o( ) 2 2 o( ) (1 2) 2 o( ):2: 8 2 o( ) C { , (C) 2 o( ):3: 8 1 2 o( m ) 2 2 o( n ) (1 2) 2 o( m+n ):$ ,1 212xlim!x = xlim!x xlim!x = 0C = C lim = 0limx!x x!x 6700000012 = lim ( 1 2 ) = lim 1 lim 2 = 0:limx!x m x!x nx!x m+n x!x m n " 1 ; 3 * &:1: o( ) + o( ) = o( ):2: o(C ) = o( ):3: o( m )o( n) = o( m+n ):= , + o( ) o( m+n ) # ./, { o( ) o(C ) o( m ) o( n) * % # 1 & (", & + ).6 o( ) O( )("O " %" ), * & (x), (x) = c 6= 1:limx!x (x) , o( ) O( ) ..
o( ) * O( ):00000$-+ ,&+1. "* *:. 4 # #* "" ; "? (c. 6.2).2. 7 #*0 * * "* ? (c. 6.1 # 6.1).3. 4 % *# # "* *: !0?(c. 6.2).4. *:, 06 (c. 6.1).5. # % "* "* "5 *: (c. 6.2{6.4).6. 1* 0 "* *:? (. 6.2, 6.3 # 6.2, 6.3).7. 4 # o (" *")? (.
6.1).68 7>8 7 8 456 & (x) (x) O_ (x0). 7.1. 8 (x) = 1lim(7:1)x!x (x) & (x) (x) + x ! x0. ": (x) (x) x ! x0: 7.1. ) (7.1) : (x) = lim 1 = 1 = 1limx!x (x) x!x (x)1 (x), , (x) (x) x ! x0:( % + .) 7.1. sin x x x ! 0, sin x = 1limx!0 x( . (3.5)). 7.2. ln(1 + x) x x ! 0, ln(1 + x) = 1limx!0x( . (4.8)).000 & 1: 8 (x) 1(x) (x) 1(x), x ! x0, (x) (x) 1(x)1(x) x ! x0:2: 8 1(x) C1 (x) 2(x) C2 (x), x ! x0, 1J 2 { C1 =6 C2 (1 (x) ; 2(x)) (C1 (x) ; C2 (x)) x ! x0.
( C1 =6 C2 1 ).69 1 : = lim ( ) = 1 1 = 1:limx!x 1 1 x!x 1 1 2 :1 ; 2 = lim 1 ; 2 = 1 lim ( 1 ; 2 ) =limx!x C1 ; C2 x!x (C1 ; C2 )C1 ; C2 x!x !11 (C 1 ; C 1) = 1:12= C ; C xlimC;C=122C1C2 C1 ; C2 112 !x 7.3. sin x ln(1 + x) x2 x ! 0 ( . 7.1 7.2). 7.4. sin 2x ; ln(1 + x) 2x ; x = x:000000 7.1 (o ./ )# # ). 1 x ! x0 * ./ , - ./ ( ) ) x ! x0 : (x) 1(x) (x) 1(x) x ! x0. = alimx!x !11xlim!x 1 = xlim!x 1 = 1 a 1 = a:.1) 8 002) 8 ( . 6.2). * 0 = 1limx!x =0limx!x 001 = 0limx!x 11 = 1:limx!x 10 6.3070$ " limx!x + & 1 1, , .3) 8 9Flimx!x 1 9Flimx!x 1, 1 " limx!x 1limx!x 1 1 1 + & .6 7.1 % " &.8 1 00000xlim!x0(x) xlim!x (x)0(x) * " , xlim!x (x)" % (x)xlim!x (x) xlim!x (x) 6= 0xlim!x (x) = xlim(x)!x", 6.4:(x) xlim!x (x) = 0 xlim!x (x) 6= 0 xlim!x (x) = 1:00000000718 * " ( % ((xx)) + 00 ), (x)limx!x (x)( 00 ) " & +, 7.1.
7.5.2 sin2 x22( x2 )2 11;cosxlim x2 = xlimx!0!0 x2 = xlim!0 x2 = 2 0 x2 ! 0 x ! 0 sin2 x2 = sin x2 sin x2 x2 x2 = ( x2 )2:? * 1 " " &, + x ! 0 :1: sin x x2: tg x x3: arcsin x x4: arctg x x2x5 : 1 ; cos x 2 6: ex ; 1 x7: ax ; 1 x ln a8: ln(1 + x) x9: loga(1 + x) lnxa 10: (1 + x)m ; 1 mx:( 1 8 " 7.1 7.2J + 5 6.5.?* 2:tg x = lim sin x = lim x = lim 1 = 1 = 1:limx!0 xx!0 x cos x x!0 x cos x x!0 cos x 172? 3 y = arcsin x. 2 = sin y. 2 y ! 0 x ! 0, arcsin x = lim y = lim y = 1 = 1:limy!0 sin y y!0 yx!0x1> 4:? 10 y = (1+ x)m ; 1, 1+ y = (1+ x)m ln(1 + y) = m ln(1 + x). 2 y ! 0 x ! 0, x ! 0ln(1 + y) y, m ln(1 + x) mx =) y mx, ..
(1 + x)m mx.6 7.1 1 % &, , , 1 * %, 1 &. 2 01+o()o()@1 +A = 1 + 0 = 1=limxlim!xx!x00 ( + o( )) x ! x0: + + o( ) = lim lim (x) = lim (x) :limx!xx!x (x)x!x + o( ) x!x (x) , % & % * " " % ( ). 7.6.22x+5x2x = 2:lim=limx!0 x + 6x3x!0 x0000< 5x2 6x3 "% , 5x2 = o(x) 6x3 = o(x) x ! 0: *, " + " " "1 .8 (x) { " & x ! x0, x ! x0:1) sin (x) (x)J2) tg (x) (x)J3) arcsin (x) (x)J4) arctg (x) (x)J732(x)5) 1 ; cos (x) 2 J6) e(x) ; 1 (x)J7) a(x) ; 1 (x) ln aJ8) ln(1 + (x)) (x)J9) loga(1 + (x)) ln(xa) J10) (1 + (x))m ; 1 m(x):' & (x) (x) x ! x0:(7:2)2! ; = lim ; = 0 1 = 0;=lim;1=0Jlimlimx!x x!xx!xx!x ..0000 ; = o() ; = o( ) x ! x0:(7:3) ", ; = o( ) x ! x0 , 01+o()o()xlim!x = xlim!x!x @1 + A = 1 + 0 = 1 =) & (7:2) = xlim, , ; = o() x ! x0 , = 1 =) & (7:2):limx!x 2 ", (7.3) ", * + (7.2).8 (x) (x) { " & x ! x0, (7.3), + " & " & % * .
) (7.3) *, 0000(x) (x) x ! x0 () = + o( ) x ! x0:74(7:4)6 (7.4) " " &, + x ! 0 * & ( *):1) sin x = x + o(x)2) tg x = x + o(x)3) arcsin x = x + o(x)4) arctg x = x + o(x)2x5) cos x = 1 ; 2 + o(x2)6) ex = 1 + x + o(x)7) ax = 1 + x ln a + o(x)8) ln(1 + x) = x + o(x)9) loga(1 + x) = lnxa + o(x)10) (1 + x)m = 1 + mx + o(x):> & * . 7.7.2x221 + x + o(x ) ; (1 ; 2 + o(x2 ))x ; cos xelim tg x2 = xlim=x!0!0x2 + o(x2)3 x2 + o(x2)3 + o(x2)22 x2 = 3 := xlim=lim!0x!0x2122 o(x ) " "% " & x ! 0 % x2.2$-+ ,&+1.
1* "* *: #0 2*? (c. 7.1).2. 1* #0 2* "* *: ? (c. 7.1 7.5).3. 4 % *# # 2* "* *:?(c. (7.3)).4. 1* #0 *? (c. (7.4) 0 ":).5. 1* #0 * ? (c. 7.11).75 8? 456( ?&& { . V & &&. (4, A*, ,%, A , " *% & { & 2) 1 &. " & y = f (x) x0. ? x0 1 Tx (* ). 2 & y = f (x) 1Ty = f (x0 + Tx) ; f (x0 ):4 %Ty = f (x0 + Tx) ; f (x0 ) :TxTx 8.1.
, % 1 Ty % 1 Tx, Tx ! 0, & f (x) x0. ( " cf 0 (x0):Ty = lim f (x0 + Tx) ; f (x0 ) :(8:1)f 0 (x0) = xlim!0 Tx x!0Tx6 " f 0 (x) " ":y0 (x) yx0 ddxy :, x = x0 " y0 (x0) f 0 (x0) ddxy jx = x0 :76 , & (8.1) * f (x) ; f (x0 ) :f 0 (x0) = xlim!x x ; x00(8:2)) "4 XOY + , y = f (x). M0(x0 f (x0)):$ M1 1 M0M1. M1 * 1 " ( . 8.1).;. 8.1 8.2.
8 M1 & M0 1 M0M1 " M1 M0, * *, , 1 + *, M0 . + . " M1 (x0 +Tx f (x0 +Tx)) ' { 1 M0M1 x. 2, a , +&& 1 M0M1 Ty :tg ' = T(8:3)x8 * M0 M1, .. Tx 0,Ty ' " 1 f 0 (x0) = xlim!0 Tx , Ty = f 0 (x ):tg = xlimtg'=lim0!0x!0 Tx77= , , 1 * x 1 M0, " J +&& tg f 0 (x0):< f 0 (x0) +&& & & f (x) M0(x0 f (x0)).< +&& M0(x0 f (x0 )), + , & f (x) M0(x0 f (x0 )):y ; f (x0 ) = f 0 (x0)(x ; x0):(8:4) 8.3. M0, M0. ( + .) : , " y = k1x + b1 y = k2x + b2 " , " , " k1k2 = ;1:(8:5)= , +&& k = ; f 0 (1x ) 0f 0 (x0) 6= 0 (kf 0 (x0) = ;1).E & &, 1 M0(x0 f (x0 )), % 1 :(8:6)y ; f (x0 ) = ; f 0 (1x ) (x ; x0):08 f 0 (x0) = 0, x = x0.
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