"Q =SDyú ? [? ?Ds [?T Z? ? tc L ?￥?!? B a 'à Q  1.  ?? ε >0? ?s?B?1 ? ? B M ? ??? TN Nmn ≥, nm xx ε?< 5? {} n x 1 cauchy' ? :T , lim ( ) 0 nm nm xx →∞ ? = b 2. ?  {} n x  ? {} k n x  k n x a→  5 ? ^ ?  {k →∞ a } n x ￥B?K?b ?? è { } (1) n ? μ 2? K?b ?  {} n x ￥Kv  Kl K? ?Ti 5?1? ? ￥   / Ki:1 lim n n x →∞  lim n n x →∞ b = a '? ?  1. ! ()x? ^ [a,b] a ê?  O ([,]) [,]ab ab? ? 5 ()x?  [a,b] i· B￥?? ?b 2. ?? ? {} n x ￥ K? /KA?· B Olim n n x →∞ " 1 sup{ , , } lim sup nn k n kn x xx + →∞ ≥ =L lim n n x →∞ " 1 inf{ , , } lim inf nn k nkn x xx + →∞ ≥ =L b 3. {} n x iK5 {} n x ￥ K?/KM?' lim n n x →∞ " lim n n x →∞ " lim n n x →∞ b 4. ? 7 uW B { }O α -? μ?A uW [a,b]' [a,b] O α α ?U 5 V { } O α ?A VGμ K? 7 uW 1 ,, n OO α α L dB-?  [a,b]b' [a,b]= 1 n OO α α UL U b ? a '1 p  . g? B? '? ? ? ?1F[V ?i ' Y ?3  Léil  2. ü ?' ? ? ^ ?D s￥ ? ? $ i ? ?¨ '? ?￡ ü > u W  ? ?f ? ￥ '?é?Btμ1 5V7g??¨'? ?é?s ?￡￥ ? ?b  1 a? ? è5  è 1 ¨? ?μ?? ?￡ ü uW*? ?'X? 1  ??μ ?? ?? ? - 1 - "Q =SDyú ? [? ?Ds [?T Z? 2  ! 1B uW* [{ nn ba , ]} à￡ []{} L,2,1,, =∈? nba nn ξ O?B ￡ ü  ￡ ü ±X/B???μ? ?  P K'1 ? p￥ ξ 1N Vüí | ?  {  } n a { } n b ?? , 1221 bbbaaa nn ≤≤≤≤≤≤≤≤ LLL yN 1?9 ?  Oμ ? è ? ??? μ?? ? i{} n a 1 b ξ= ∞→ n n alim  O  L,2,1, =ξ≤ na n ?y () nnnn aabb +?= 7 ( ) 0lim =? ∞→ nn n ab # ξ=ξ+=+?= ∞→∞→∞→ 0lim)(limlim n n nn n n n aabb  Oy ?hA P{} n b ξ≥ n b ?ü￡¤ [ ]{ } L,2,1,, =∈ nba nn ξ  Ka ¨Q￡E￡ ü ?N￥ ξ?B Y L   ? 6μB? [ ]{ } L,2,1,, ' =∈ nba nn ξ  5? )(0)( ∞→→?≤ξ′?ξ nab nn  ?áD 0 ' >?ξξ M ±  è 2 ¨ ? ?? ?￡ ü uW*? ?'X? 1  ?? ? ?? ? d bμ ?￥ ?"Aμ  ??  2  ! 1B uW* [{ nn ba , ]} à￡i?B￥? []{} L,2,1,, =∈ nba nn ξ  ￡ ü ￡ ü ±Xó B ?" μ ? P¤ ￥  ??'1 ? p￥S S ξ 1N | { } n aS =   ?i è ? ? ??? ?i 1 b {} n asup=ξ  n5 ? ξ1 { } n a ￥B? ? # L,2,1, =≤ na n ξ ? ξ ^ { } n a ￥Kl ?  μ ? ξ< m b 5 m b ?? ^ ￥  ?'{} n a mk ba >? ? D [ ]{ } nn ba , 1 u W* M ± ( ) ji ba , b ?[ ?? ξ≥ n b ?ü￡¤ L,2,1, =≤ξ≤ nba nn  1? ξ￥?B?D è 1?￥￡ üM] è 3  ! ^> uW [ ￥?9f ? ??A ?? ?T 5 P ()xf ] ] ba, () () ,, bbfaaf ≤≥ [bax , 0 ∈? ( ) 00 xxf =  ?ê vDù? 3 ?D k5  - 2 - "Q =SDyú ? [? ?Ds [?T Z? ￡EB  ¨ ??/ ??<>1  è ￡E   !"? (){}bxaxxfxF ≤≤≥= , FFa ,5 ∈ ? b [ ] FbaF ,,? μ?? ?? e ? μ  ?? ! 5F Fx sup 0 = [ ]bax , 0 = /￡ ( ) 00 xxf =  &? ? μ ?Fx ∈ 0 () 00 xxf ≥ ( ) ( ) bbfxf ≤≤ 0 ¤ ( ) [baxf , 0 ]∈ ? ?9? μ()xf () 00 xxf ≥ ( )()( ) 00 xfxff ≥  Vn ( ) Fxf ∈ 0 ?  ? ^ o ? μ Fx sup 0 = ? () 00 xxf ≤ () 00 xxf =  &a ? 5 i =￥ ? Fx ? 0 F { } n x  P  0 xx n ↑ ( )∞→n 9i ?   (){} n t btx n ≤< 0 0 xt n ↓ ∞→n ? ?9 f Fx n ∈ [# Ft n ? üμ T  ?? ? ? 7() () () nnnn ttfxfxfx <≤≤≤ 0 n ∞→n ¤ () 000 xxfx ≤≤ ? ^μ () 00 xxf = ￡E=  ¨ uW*/ ? ?<> 1 è ￡ E  ? ( ) aaf =  H  bü ^Z? () bbf = a () xxf =  ￥ L?[/9 ![ba, ] ( ) aaf >  ( ) bbf < s u W  !s?1 μ ü ^Z?[ba, ] c () ccf = c ( ) xxf =  [ ]ba, ￥ L? 1? óe 9 [/9 !??C?? f ?  ? ( ) ccf >  | bbca == 11 ,  ?  |  ?N¤B ) uW [ G N/ uW* () ccf < cbaa = 1 = 1 , ] 11 ,ba [ ]{ } nn ba ,  μ  ? uW *? ? n? () nn aaf > () nn bbf < 0 x?  P  ?? μn [ ] nn bax , 0 ∈ C￡  Y L  ?i?() 00 xxf = ∞→n H ? [# ?9 üμ 0 xa n ↑ 0 xb n ↓ f  () () ( ) nnnn bbfxfafa <≤≤≤ 0  7 ¤ ? ^μ∞→n () 000 xxfx ≤≤ ( ) 00 xxf =  - 3 - "Q =SDyú ? [? ?Ds [?T Z? è 4  ! > uW [ f ?]ba, ( )xf ?? ( )xg ?9  Oμ ( )()()(bgbfagaf > )< ,  k￡ üZ ?  uW() ()xgxf = ( )ba, =μ L?   = v  M ? Vù? 3 ? D k5 ￡ ü / uW* P[]{} nn ba , ( ) ( ) ( ) ( ) nnnn bgbfagaf >< , ? uW*? ?  ξ?  P μn? [ nn ba ,∈ ]ξ C￡ ( ) ( )ξξ gf =  Y L  ? ( )xg  [ ]ba, ￥?9?? ￥/[# ? μ [ nn ba , ] ξ↑ n a ξ↓ n b  () ( )()( ) ( ) nnnn bfbggagaf <≤≤< ξ  ?i? ?()xf ??? )FJOFBie5 μ  () () ( ) ( )ξξ fbffaf n n n n == ∞→∞→ lim,lim  ? () () ()ξξξ fgf ≤≤ ? () ( )ξξ gf = ξ1Z? ( ) ( )xgxf =  uW ( =￥ L?) ] ba, è 5 k￡ ü  uW [ ￥ ?8 L ? ^? V ￥1,0 ￡ ü  ¨ u W*/ ?  8¨Q￡E Q ! u W [ ]1,0 ￥ ?8 L ? ^ V ￥ ' V ??B   LL ,,,, 21 n xxx ü uW [ ??s ?¤ ?? uW?à μB? uW?c :? uW1B) uW ü uW ??s ?¤ ?? uW?à μB? uW?c :? uW1=) uW [ GN ¤ uW*  ? uW ] ] ] ] ]} 1,0 1 x [ 11 ,ba [ 11 ,ba 2 x 22 ,ba [{ nn ba , [ ] nn ba , ?c ? uW* ? ?  n xxx ,,, 21 L ξ?   P μn? [ nn ba ,∈ ]ξ ? ?μ ? n? μ [ ] nn bax , 0 ? 7 [] nn ba ,∈ξ  ξ≠? n x  ±   - 4 - "Q =SDyú ? [? ?Ds [?T Z? ? ' è ? ú ?T￥￡ ü153s?￥￡ ü÷F b# ?aˉ 5  1 p ?  {Jn}￥ a/ ?? (1) 1 1; n x n =? (2) [2 ( 2) ]; n n xn=+? (3) 221 1 ,1(1,23, kk xkx k k + = =+ = L ); (4) 1 [1 ( 1) ] ; n n n x n + =+? (5) (1) 12 ; n n n n x ? =+ (6) 12 cos . 13 n nn x n π? = + 2 !  ?l p￡ ()fx D (1) sup{ ( )} inf ( ); xD xD f xf ∈ ∈ ?=?x (2) inf{ ()} sup (). xD xD f xf ∈ ∈ ?=?x 3 ! sup Eβ =  O Eβ?  k￡1 E ? Vê | ?  {} n x O n x o?M] P lim n x x β →∞ =  ? ? Eβ∈ 5 f? ?? ? 4 k￡ l ? ? A μ  ??? / ? ? t ? +∞￥ ? A μ/ ?? t? ￥ ?  Aμ  ?? ?∞ 5 ksY   ?@/ Hq￥ ?  (1) μ  ??í/ ??￥ ?  (2) cμ  ????cμ/ ??￥ ?  (3) ;c μ  ???cμ/ ??￥ ?  (4) ;?cμ  ????cμ/ ??￥ ?  ? a/ ???μK 6 ?¨μK -?? ? 92￡ ü?á? ? ? 94 7 ?¨?á ?? ?￡ ü??μ? ? AμK 8¨ uW* ? ?￡ ü??μ? ? AμK 9 ks uW*? ?￥Hq ?|> uW ?1 7 uW 2T8" ? ?|Hq ??|Hq 11 2 2 [,] [, ]ab ab??L 0 nn ba? → ??2T8" ? k  è a ü - 5 - "Q =SDyú ? [? ?Ds [?T Z? 10 ? {} n x í? Odí kv  5Ai ?0  , kk nm x xa→∞ → (a1μK ? ) 11μ? ?  {} n x ?? l ?5Ai ?0  ,) kk nm x ax b b→→ (α≠ 12 p￡ ?  { μ?￥ 1Hq ^ { ￥ ??0 ?  { ?μ l ?￥0 ?  } n a } n a } k n a 13 !  [, ?l O ?B? )f ?￥ Ki  p ￡  [,  μ? ()fx ]ab ()fx ]ab 14 !  í?  p ￡i ()fx [,]ab [,]cab∈  ? ó 0δ > f ? ()fx (, )[,cc ab]δ δ?+∩ í? 15 ! ^ (, ￥jf ? Oμ ? p￡()fx )ab lim ( ), lim ( ) xa xb f xf +? →→ x i 16 !  [, oμ?B ?W??l ()fx ]ab () | ( 0) ( 0)|.xfx fxω = +? ? p￡ ?i 0, ( )xε ω> ≥ε ) ￥? oμμK? x 17 !   ?? Oμ? ?i()fx [0, )+∞ (,a∈ ?∞ +∞   [0  oμμK??í? p￡ ()fx a= , )+∞ lim ( ) x f x →+∞ i 18 !  (, ?? p￡  (, Bá ??￥ 1Hq ^ ()fx )ab ()fx )ab lim ( ) xa f x + → D lim ( ) xb f x ? → ?i 19 p￡ ?  11 1 2 n x n =+ + +L ? H￥K?i n→∞ 20 ?¨ O  l ?? ?) ?/  ? ￥ l ?? (1) 01 2 (| | 1,| | ); n nnk x aaqaq aqq a M=+ + ++ < ≤L (2) 2 sin1 sin 2 sin 1; 22 2 n n n x =+ + + +L (3) 1 11 1 1( 23 n n x n + =? ++ +?L ). 21￡ ü 0 lim ( ) xx f x → i￥ 1Hq ^  ?i ó? 0ε > i 0δ > ? 00 0|'|,0|' |xx xxδ δ<?< <?< H?μ - 6 - "Q =SDyú ? [? ?Ds [?T Z? | ( ') ( '') | .fx fx ε? < 22￡ ü ()fx 0 x ? ?? ￥ 1H q ^ ? ó 0ε > i 0δ > ? 00 0|'|,0|' |xx xxδ δ<?< <?< H?μ | ( ') ( '') | .fx fx ε? < 23￡ ü/ K?i (1) 12 cos ; 13 n nn x n π? = + (2) (1) 12 ; n n n n x ? =+ (3) 2 sin( ); n x nnπ=+ (4) cos ; n x n= (5) tan . n x n= 24 !   V? |' ??/?  O()fx (, )a +∞ ()fx| lim ( ) x f x →+∞ i  p￡  lim '( ) 0 x xf x →+∞ = 25 !  (, V? O |'()fx )?∞ +∞ ()| 1fx k≤ <  ?ó 0 x  7 1 ()( 0,1,2,), nn xfxn + = = L p￡ (1) lim n x x →∞ i (2)  ?K1 ()x fx= ￥? O ^·B￥ 26 !  [, ?@Hq ()fx ]ab (1) |( ) ()| | |, , [,], 1;fx fy kx y xy ab k?≤? ?∈ 0<< (2) ￥′×c [, = ()fx ]ab 5 ?i 0 [,]x ab∈  7 μ 1 ()( 0,1,2,) nn xfxn + ==L (1) lim n x x →∞ i (2)Z? ()x fx= ￥3 [,  ^·B￥??3ü ^  ?K′ ]ab - 7 - "Q =SDyú ? [? ?Ds [?T Z? 27 !   ??i O Kv′?()fx [,]ab 0 x ^·B￥? ! 0 [,]x ab∈  P 0 lim ( ) ( ) n x f xfx →∞ =  p￡ 0 lim n x x x →∞ = 28 !  [,  ?? V±? ! ()fx ]ab (1) min ( ) max ( ); axb axb f xp fx ≤≤ ≤≤ < < (2) ?T ()fx p= 5μ '( ) 0fx≠  p￡ ￥?oμμK? ()fx p= 29 !  [, ??()fx ]ab () 0fa<   p￡ i() 0fb> (,)abξ∈  P () 0f ξ =  O () 0( )fx x bξ><≤ 30 ! ^ [, ￥ ??f ? Kv′?Kl′sY1()fx ]ab M ?  p￡ Ai uW [, (mm M< ) ]α β  ?@Hq (1) () , ()fMfmα β= = () , ()fmfMα β= =  (2) ?()mfxM<< (, )x α β∈  31  [0 ?? O()fx ,2 ]a (0) (2 )ffa=  p￡ i [0, ]x a∈  P  () ( )fx fx a=+ 32 !  [,  ?? O |′1? ? p￡()fx ]ab ()fx≡è ? 33 !  (, Bá ??()fx )ab ,ab≠±∞￡ ü  μ? ()fx (,)ab 34 ?f ?  (,  ?@ ? ?G (Lipschitz)Hq'iè ?()fx )ab K P¤ | ( ') ( '') | | ' '' |, ', '' ( , ).fx fx Kx x xx ab?≤? ∈ ￡ ü  (, Bá ?? ()fx )ab 35 k¨Bá ??￥?l￡ ü ?f ?  [, ? [, ?Bá ?? 5  9Bá ?? ()fx ]ac ]cb ()fx [,]ab 36 !   ?? O()fx (,?∞ +∞) lim ( ) x f x →?∞ D lim ( ) x f x →+∞ i￡ ü  ()fx - 8 - "Q =SDyú ? [? ?Ds [?T Z? (,?∞ +∞) Bá ?? 37 ?  u W()fx X (μ k í k )?  μ μ?￥ ? ? ' |' 5  ()| ,fx MxX≤ ∈ ()fx X ?Bá ?? 38 p￡ () lnf xx= x (0, )+∞ Bá ?? 39 !  (,  V? O()fx )a +∞ lim '( ) x fx →+∞ =+∞ p ￡   (, ?Bá ?? ()fx )a +∞ 40 p￡ () lnfx x x= (0, )+∞ ?Bá ?? 41 ?/ f ? uW [0 ￥ V? ,1] (1)  [0 μ?? ???1()fx ,1] 1 (1,2,xn n = = L ) (2) sgn(sin ), (0,1], () 0, 0; x fx x x π ? ∈ ? = ? ? = ? (3) 11 ,(0,1 () 0, 0; x fx xx x ? ?? ? ∈ ? ?? = ??? ? = ? ], (4) 1 ,(0,1 1 () 0, 0. x fx x x ? ∈ ? ?? ? = ? ?? ?? ? ? = ? ], 42) ? 2 (), (),| ()|f xfx fx ??W V?￥1" 43 ! (), ()fxgx ? [,  V￡ ü ]ab () max( (), ()), () min( (), ())M x fxgx mx fxgx= =  [, 9 ^ V￥ ]ab 44 !  [,  V O  p￡ ()fx ]ab () 0fx r≥> (1) 1 ()f x  [, V ]ab - 9 - "Q =SDyú ? [? ?Ds [?T Z? (2)  [, V ln ( )fx ]ab 45 !  V p￡ ?ó()fx [,]ab 0ε > i ? 1è ? ￥f ? ()x?  P |() ()| . b a fx x dx? ε?< ∫ 46 !  [, μ??l ()fx ]ab [,] [,] [,] sup () inf (), f xab xab ab f x f xω ∈ ∈ = ? p￡ ', '' [ , ] [ , ] sup | ( ') ( '') | . f xx ab ab f x f xω ∈ = ? 47 ! ()fx 0 x ?íμ?l Oμ??l 000 11 () lim , f n xxx nn ω →+∞ ?? =?+ ?? ?? . p￡ ()fx 0 x ??￥ sA1Hq1 0 ()0 f xω =  48 ?f ?  [,()fx ]A B V￡ ü 0 lim | ( ) ( ) | 0, b ah fx h fx dx → + ?= ∫ ? (?B?é?1s￥ ??? ) AabB<<< 49  ?i 8￡( ) 0, ''( ) 0,fx f x≥ ≤ [,]x ab∈ ? ? p￡ 2 () () . b a f xf ba ≤ xd ? ∫ 50 !  [, μ ??￥?f ? p￡ ()fx ]ab 1 max | ( ) | | ( ) | | '( ) | . bb aaaxb f xfxdfx ba ≤≤ ≤+dx ? ∫∫ 51 !  [, V p￡ i ??f ?? ()fx ]ab (), 1,2, n xn? = L  P lim ( ) ( ) . bb n aan x dx f x dx? →∞ = ∫∫ 52 !  [, ó ￡ V p￡ ()fx ]ab (1) i  uW?  {[ P , ]}ab 11 [,](,)(,) nn nn ab ab ab ++ ??, O 1 ([ , ]) fnn ab n ω <  - 10 - "Q =SDyú ? [? ?Ds [?T Z? (2) i  P¤  ? ?? 1 [,] nn n cab ∞ = ∈ I ()fx c (3)  [, μí k? ??? ()fx ]ab - 11 -