1. Математический анализ (850924), страница 11
Текст из файла (страница 11)
> 4.10. N ( ( )% 0:( 2x < 0,f (x) = x4 ;+x4 x > 0.< D% * f (0 + 0) = f (0 ; 0) = 4 %%% $) f (0) , (, <%=+ f (x) )% = 0 ! $ ($ I- ). > 4.11. N ( ( )% = 0:( 2x < 0,f (x) = x3 ;+x4 x > 0.2 + 4) = 4< x limf(x)=lim(3;x)=3limf(x)=lim(x0+0x 0x 0 0x 0f (0 + 0) 6= f (0 ; 0): C), = 0 { )% $ %)%)% <%= f (x): > 4.12. N ( ,% $ <%= f (x) = 21=x )% = 0:< ? $ y = x1 )lim 21=x = y lim2y = 1+x 0+0x 0 0! ;!!!! ;!!!11=x = lim 2y = 0:lim2yx 0 0% %% $ , f (0+0) = 1 )% = 0{ )% $ II- .
>180! ;!;1 4.13. N ( ,% $ <%=1=x ; 12f (x) = 21=x + 1 )% = 0:< ;! f (0 ; 0) f (0 + 0) :1=x ; 10 ; 1 = ;12=limf(x)=limx 0 0x 0 0 21=x + 10+11=x ; 11=x (1 ; 2 1=x)22lim f (x) = x lim= lim=x 0+00+0 21=x + 1 x 0+0 21=x (1 + 2 1=x)1=x1 ; 0 = 1:1;2== x lim0+0 1 + 2 1=x1+0C ( ($ <%, )1!11=x= 1 = 0:lim 2 = x limx 0+00+0 21=xN%,f (0 ; 0) = ;1 6= f (0 + 0) = 1:? (, )% = 0 { )% $ %) %)%()% $ I- ). >! ;! ;;!!;!;;!;!!(' 2" ! "$ &'"" " = 0: ; sin x x 0, ; sin xx 0,1) f (x) = cos x ; 1 x 03) f (x) = 2 cos x ; 1 x > 0 3 sin x x < 0,1 :2) f (x) = ;4)f(x)=cos x ; 1 x > 01 + 2 =x11) 7 "3) 6" " 2) I" " !" "4) 6" " .||||{181 5 ( + " ++ = x0 $+ <%<%= f (x) ,+ ' $ , f (x0 0) = +1 ; 1:"%( + <% <%= $ )1 + $)! '$)% $ <%= II- .)*1 "* 5.1.
;! ( %$ <% <%=3;4xf (x) = x2 :< M%=+ f (x) , % )% = 0: N <%= % !)%: ;4 ! ;4 !f (0 ; 0) = +0 = ;1 f (0 + 0) = +0 = ;1:% '$, ++ = 0 +++ %(! ! <% <%= f (x): [% '1+ <% % %$ . 5.1.;. 5.1"+, ( %+ $(+ <%:3;4xf(x)k = xlim x = xlim x3 = 1182!1!10 31x;4b = xlim (f (x) ; kx) = xlim @ x2 ; xA = xlim ;x24 = 0:!1!1!19' * %), $), (:y=x{ %! .? $(3;4xy ; y = x2 ; x = ; x42 :% %% y ; y < 0 + 8x 6= 0 <% <%= 1 1%! .( 1 ( %$ <% <%= (. 5.2). >;. 5.2 5.2. ;! ( %$ <% <%=3xf (x) = x2 ; 4 :< )% $ <%= 1 = ;2 2 = 2: N <%= '$ , )%: ;8 ! ;8 !f (;2 ; 0) = +0 = ;1 f (;2 + 0) = ;0 = +18!8!f (2 ; 0) = ;0 = ;1 f (2 + 0) = +0 = +1:9 , ) <% <%= f (x) ( %( x = ;2 x = 2:183;[% '1+ <% % %$ .5.3..
5.3"+, ( %+ :2f(x)xk = xlim x = xlim x2 ; 4 = 10 31xb = xlim (f (x) ; kx) = xlim @ x2 ; 4 ; xA = xlim x24;x 4 = 0:9' * %), (, * (%+) !1!1!1!1!1y = x:? $( y ; y )3xy ; y = x2 ; 4 ; x = x24;x 4 :6 1 '(# + > 2 (# + < ;2 $), ) <% <%= + % , ! +1 $ ! ;1: :)+ %$ )+, ) f (0) = 0 %$ <% (.5.4). >184;. 5.4(' 2" ! "$ &'"<" " F '" 2) f (x) = 5xx+ 1 :1) f (x) = x 1; 1 4;; 231)2)||||{185 69 + . 1 9 + 0 M%=+ (x) $+ & ! x0, xlimx (x) = 0:.
<%= () () { '%) ! x0, , ,$( $ (x) (x) (x) (x)%1 +++ '%) ! 0.M%=+ f (x) $+ # x0, 9 > 0 > 0 : 8x 2 _(x0) jf (x)j .. <%=+ () { '%) + ! x0, <%=+ f (x){ )+ % )% x0, $ (x) f (x)+++ '%) ! <%=! ! x0. 6.1. @%$(, ) <%=+(2x 6= 1,f (x) = (10x 9; 1) x = 1+++ '%) ! ! 1.< @!(,limf (x) = lim(x ; 1)2 = 0:x!x!!1x6=101x6=16 $), ) <%=+ f (x) +++ '%) ! x ! 1, + , ) $) f (1) = 109 ((%). > 6.2. ;( %$ <% <%= = () { '%) ! ($*!) ! 3 ), % 1) (3) = 02) (3) = 43) (3) :186; ;;< 6%$ <% %,.1)3)2)> 6.3. ;!1:limxsinx 0x!< C, ) xlim0 sin x1 * ($+ ( $ +.
; % %% sin x1 <%=+ !1)+ j sin x j 1 8x 6= 0 , <%=+ x { '%) + x ! 0, $ '%) ! <%= ) x sin x1 +++ '%) ! <%=! ! 0:1 = 0: >limxsinx 0x!!(' 2" ! "$ &'"<" xlim x arctg x1 :: 0.2!0||||{187, ( ( (x) () { '%) <%= ! x0. .(x) = 0limx x (x) <%=+ (x) $+ & ,# (x). .(x) = C 6= 0 1limx x (x) <%= (x) (x) $+ & # . " ), (x) = 1limx x (x) % '%) $ 0 x ! x0 #(x) (x) x ! x0: ! 0sin a ; 1 ln a(1+ )m ; 1 mpntg e ; 1 p 1 + ; 1 =narcsin loga(1 + ) = ln a 1 + ; 1 =2arctg ln(1 + ) 21 ; cos =2:!!0!00)*1 !"# -* (* 6% '%) <%= ($+ ) #! , '%) , ( + %+ ! (0=0)) : #+ , '%) , x ! x0 $+, %1 ( (% $ ,) $( % '%) x ! x0.
6.4. ;!x :limx 0 sin 2x188!< 2x ! 0 x ! 0 , $), sin 2x 2. C++ $( % '%) , )0!xx = 1: >lim==limx 0 sin 2x0 x 0 2x 2 6.5. ;!sin 2x :limx 0 tg 3x!2x = 0 = lim 2x = 2 : >< xlim0 sintg 3x0 x 0 3x 3 6.6. ;!sin 3x :limx sin 2x< C, ) 1 $ ($(+ '=!%, % %% 3x 6! 0 2x 6! 0 x ! , '$ '(, $ y = x ; % %% x ! , y ! 0. y = x ; !sin 3x = 0 = x = y + = lim sin 3(y + ) =limx sin 2x0 y ! 0 y 0 sin 2(y + )!!!!!!!! 3y !;sin3y3: >sin(3y+3)= lim=lim=lim;=;y 0y 0 sin(2y + 2 ) y 0 sin 2y2y2!!! 6.7.
;!21;xlim:x 1 sin x!y=x;1221;x01;(y+1)< xlim1 sin x = 0 = x = y + 1 = ylim0 sin (y + 1) = y ! 0 2 ; 2y;yy(y + 2) = lim y(y + 2) = 2 : >= lim=limy 0 sin(y + ) y 0 sin yy 0 y!!! 6.8. ;!!!!lim(x ; 2) ctg x:x 2!189< % %% xlim2(x ; 2) = 0 xlim2 ctg x = 1 ( 0 ! 0 1, % 1 '$( ( y = x ; 2 !0 .(x ; 2) = 0 = x = y + 2 =lim(x;2)ctgx=(01)=limx 2x 2 tg x0 y ! 0 !!!!yy = lim y = 1 : >y=lim=lim= limy 0 tg (y + 2) y 0 tg(y + 2 ) y 0 tg y y 0 y!!!! 6.9. ;!p2x + 9 ; 3:plimx 0 x2 + 1 ; 1qp20!2;131+(x=3)x+9;3== lim p=< xlim0 p 2x +1;1 0 x 01 + x2 ; 12= lim 3 (1=2)(x=3) = 1 : >!!!(1=2)x2 6.10.
;!3x 0!ln(1 + 2x ) :limx 0x2!22)02xln(1+2x= 0 = xlim0 x2 = 2: >< xlim0x2 6.11. ;!log2(3 + x) ; log2 3 :limx 0x!< xlim0 log2(3 + xx) ; log2 3 = 00 = xlim0 log2 (1x+ x=3) == xlim0 x=3xln 2 = 3 ln1 2 : > 6.12. ;!e xp; 1 :limx 0 2 x2!!!!!!!ppx 1x ; 1 0!e< xlim0 2px = 0 = xlim0 2px = 2 : >190!p!! 6.13. ;!x; :limx e ; ex y = x ; ! = 0 = x = y + = lim y =< xlim ex ;; ex 0 y ! 0 y 0 e ; ey+= ylim0 ;e (eyy ; 1) = ylim0 ;ey y = ;e : > 6.14. ;!1 ; cos(1 ; cos x) :limx 0x4< % %% (x) = 1 ; cos x { '%) + <%=+ x ! 0, !!!;!!!2 =2 22 (x) (1 ; cos x)2xx4 :1;cos(1;cos x) = 1;cos (x) 2 ==228? (,4 =8 1x1;cos(1;cosx)= xlim0 x4 = 8 : >limx 0x4 6.15. ;!ln cos x :limx 0x20!lncosxx ; 1)) =< xlim0 x2 = 0 = xlim0 ln(1 + (cosx22 =2);(xcosx;1= j(cos x ; 1) ! 0 x ! 0j = xlim0 x2 = xlim0 x2 = ; 21 : >!!!!!!!? 1 : (x) (x) (x) (x) x ! x0, (x) (x) (x) (x) x ! x0:" 1 + ) '! '%)! 1( 1 $+( %!.1910000 6.16.
;!x tg 3x :limx 0 1 ; cos2 2xtg 3x = limx 3x< xlim0 1 ;x cos2 2x x 0 (1 ; cos 2x)(1 + cos 2x) =23x= xlim0 ((2x)2=2) (1 + cos 2x) = 2(1 +3cos 0) = 34 : >9' , ) ( '%) +( % ) #+ '*) ($+. %+ $ 1 % ( $(. 6.17. ;!tg x ; sin x :limx 0x3< . ($( % tg x x sin x x x ! 0, )x ; x = lim 0 = lim 0 = 0limx 0 x3x 0 x3 x 0 %% 2 =2)tgx(1;cosx)x(x1: >tgx;sinx=lim=lim=limx 0x 0x 0x3x3x32" ', )+, + %+ 1 ($( ,. 6.18. ;!2x ; arctg 3x :limx 0 2x + arctg 3x; arctg 3x = lim 2 ; (arctg 3x=x) = 2 ; xlim0(arctg 3x=x) =< xlim0 22xx +arctg 3x x 0 2 + (arctg 3x=x) 2 + xlim0 (arctg 3x=x)!!!!!!!!!!!!!!!!2 ; xlim0 (3x=x) 2 ; 3= 2 + lim (3x=x) = 2 + 3 = ; 15 : >x 0!!1929 ( '%) 1 +( % ! * 1 +: (x) C1 (x) (x) C2 (x) x ! x0 C1 (x) 6= C2 (x) x 6= x0 ((x) ; (x)) C1 (x) ; C2 (x) x ! x0: 6.19.
;!cos 5x ; cos 2x :limx 0x2< xlim0 cos 5x x;2 cos 2x = xlim0 (cos 5x ; 1) x+2 (1 ; cos 2x) = (cos 5x ; 1) ; 25x2 (1 ; cos 2x) 4x2 = 2x2 x ! 0 2 22= =25x2; 2 6= ;2x x 6= 02 =2) + 2x225 + 2 = ; 21 : >;(25x=;= xlim0x222 6.20. ;!ex ; cos x :limx 0 x sin x!!!!2!x ; cos xx ; 1) + (1 ; cos x)e(e=< xlim0 x sin x = xlim0x sin x ex ; 1 x2 1 ; cos x x2 x ! 0 2= =2x2x 6= 2 x 6= 022= xlim0 x +x(2x =2) = 1 + 21 = 32 : >22!!2! %, '%) , 1 %'( )+ . 6.21. ;!pcos x1;:limx 0 sin 3x193!< ++ ,% 1+ +1 1,)pcos x 0 !1;1 ; cospxlim==limx 0 sin 3x0 x 0 sin 3x(1 + cos x) =2 =2x= xlim0 3x(1 + pcos x) = xlim0 6 x 2 = 0: >!!!! 6.22. ;! 1!1lim;:x 0 sin x tg x 1!0!1tgx;sinx< xlim0 sin x ; tg x = (1 ; 1) = xlim0 sin x tg x = 0 =x(1 ; cos x) = lim x2=2 = 0: >= xlim0 tg sinx 0 xx tg x!!!!!(' 2" ! "$ &'" :1) xlim tg kx x2) xlim 1 ;xcos x cos x 3) xlim 1x;sin2x4) xlim 1 ;sinx x= !0!023!0!22 5) xlim=2 ; x tg x1 ; sin(x=2)6) xlim cos(x=2) (cos(x=4);sin(x=4))pxx;7) xlim px ; 1 8) xlime lnxx;;e1 :!2!2!1!1) k2) 12 3) 43 4) 2 5) 1||||{194p6) 22 7) 38) 1e : 7( (:,.
; 'x0,( <%=+ y = f (x) %! % )%2x = x ; x0 { * <%= * $)% x0 )% x,2y = f (x0 + 2x) ; f (x0 ) { * * <%=. <%= y = f (x) x )% x0 $+ )2y :f (x0) = xlim0 f (x0 + 22xx) ; f (x0) %% y = xlim0 2x 7.1. ;! f (0), 8>1< 2f (x) = >: x sin x x 6= 00 x = 0:< 2y = f (0 + 2x) ; f (0) = f (2x) ; 0 = 2x2 sin 21x 2 x 6= 0:2 sin(1=2x)1:2x=lim2xsinf (0) = xlim0x 02x2x% %% 2x { '%) + 2x ! 0, sin 21x { )+ <%=+ % )% 2x = 0, $ 2x sin 21x +++ '%) ! <%=! 2x ! 0 f (0) = 0: >00!!00!!0(' 2" ! "$ &'"<" f (0), 8< ln(1 + 2x ) x 6= 01) f (x) = :x0 x = 0081<2) f (x) = : x cos x x 6= 00 x = 0:321951) f (0) = 22) f (0) 1.00||||{( '* 9% $ ! <%= $+ . <<= <%= ($+ '= $ ,(xn) = nxn 1(arcsin x) = p 1 2 1;x(ax) = ax ln a(arccos x) = ; p 1 2 1;x(ex) = ex(arctg x) = 1 +1 x2 (loga x) = x ln1 a (arcctg x) = ; 1 +1 x2 (sh x) = ch x(ln x) = x1 (sin x) = cos x(ch x) = sh x(cos x) = ; sin x(th x) = 12 ch x1(tg x) = cos2 x (cth x) = ; 12sh x1(ctg x) = ; sin2 x <<=+: C { ++, u = '(x)v = (x) { <%=, * $ , (C ) = 0(uv!) = u v + uv u = u v ; uv (v 6= 0):(Cu) = Cu vv2(u v) = u v 7.2.
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