"Q =SDyú ? [? ?Ds [?T Z? ? Bc±s?′? ?# ?¨ B a 'à Q  1. ?f ?  u Wf I μ?l 0 x I∈ b ?i 0 x ￥ #×  P¤? ? i ￥ 0 ()Ux 0 ()x Ux∈ μ 0 () ()f xfx≥ 5 ? ?f 0 x |¤v′ ?? 0 x 1v′?b ?i  0 x ￥ #×  P¤? ?i￥ 0 ()Ux 0 ()x Ux∈ μ 0 () ()f xfx≤ 5 ? ?f 0 x |¤l′ ?? 0 x 1l′?b v′al′d?1′v′?al′?d?1′?b 2. !f ? 1?l uWf I ￥f ? ? I  ?i ? 1 x a 2 x ? ?i L ? (0,1)λ∈ 9μ 121 ((1))()(1)( 2 )f x x fx fxλ λλ λ+? ≤ +? 5? 1f I ￥jf ?bQ- ?T9 μ 121 ((1))()(1)( 2 )f x x fx fxλ λλ λ+? ≥ +? 5? 1f I ￥?f ?b 3. ! wL ? (()xfy = 0 ,() 0 x fx ))μ,V wL￥ ML O M?í ? wL ML ￥ §sY ^?ìj?ì?￥? H? ( 00 ,()x fx )1 w L ( )xfy = ￥.?b = a '? ?  1. ?f ? f ?@[/Hq  1 f [ ]ba,  ??  2 f ( )ba, = V?b 5 () =à iB? ba, ξ P¤ () () () f bfa f ba ξ ? ′ = ? b 2. ? ?@ ?/Hq  1f [ ]baf ,∈   2 f ( )ba, = V?  3  5i () ()bfaf = ξ ∈(ba, ) P¤ () 0f ξ′ = b 3. ?f ? f () ( ) [ ]bauufyugxg ,,, ∈==  ?@ ?/Hq  1   2 g ( = V?  3 à μB??1 0  4 ,[,fg ab∈ ] )f ba, ,fg′′ ( ) ( )bgag ≠ bi ξ ∈()ba,  P¤ () () () () () () f fb fa ggbga ξ ξ ′ ? = ?′ b 4. ?f ?? 0 x ￥ #× = μ?l O  ? 0 x V?b ? 0 x 1 ￥′ ?51μf - 1 - "Q =SDyú ? [? ?Ds [?T Z? 0 ()0fx′ = b ' V? ′?￥? ?1 ,b +?il V?′?￥ ML ü?? à  ? ?@Z? x 0 ()0fx′ = ￥?1×??b 5. ?f ? f [ ]ba,  V? O () ()f afb +? ′ ′≠ k1o? ()f a + ′ ? ()f b ? ′ -W￥ ?B L ?5à iB? (,)abξ∈  P¤ ()fkξ′ = b 6. ! ) uW()xf I  V?5 ( )xf  I ?9 h ( ) 0( 0)fx′? ≥≤ 7. ?f ? f ( )ba, = V?5 f ( )ba, =?ì?9 h￥ 1Hq ^  &? B M (,)x ab∈ μ   &a() 0( 0)fx′ ≥≤ ( )ba, =￥ ??0 uW  () 0fx′ ≠ b 8. ?f ? f [ ]ba, i°? n¨￥ ??? f ?  ( )ba, =i n 1¨? f ? 5  ?ió?￥ 0 ,[,]x xab∈ à iB? (,)abξ∈ P¤ () (1) 100 00 0 () () () ()() () () () 1! ! ( 1)! n n nn fx f x f fx fx xx xx xx nn ξ + + ′ =+ ?++ ?+ ? + L 0 9. ! ?f 0 x ??  #× U( 0 x ,δ ) = V?  1 ? ? 0 (, 0 )x xxδ∈? H 0 ()0fx′ ≤  ? 00 (, )xxxδ∈ + H 5  ? 0 ()0fx′ ≥ f 0 x |¤Kl′  2 ? ? 0 (, 0 )x xxδ∈? H 0 ()0fx′ ≥  ? 00 (, )xxxδ∈ + H 5  ? 0 ()0fx′ ≤ f 0 x |¤Kv′  3 ? ()fx′ 00 (,)x xδ? 00 (, )xx =??|5? 0 x ? ^′?b δ+? 10. ! ?f 0 x ￥  #× U( 0 x ,δ ) =B¨ V?  x" 0 x )=¨ V? O  5μ  1 ? 0 ()0fx′ = 0 ()0fx′′ ≠ 0 ()0fx′′ < 5 f 0 x  |¤v′  2 ? 5  0 ()0fx′′ > f 0 x  |¤l′b 11. ! f 0 x ￥  #× =i°? n1¨ ?f ?  0 x ) n¨ V? O   k" 12ln1 5 1? n1 } ? H f  () 0 ()0 k fx= () 0 ()0 n fx≠ 0 x |¤′ O? H |v′ ? H |l′  2? n1  ? H  () 0 ()0 n fx< () 0 ()0 n fx> f 0 x ? |¤′b - 2 - "Q =SDyú ? [? ?Ds [?T Z? 12. ! 1 uWf I ￥ V?f ?5/ ? ?oM?N  1 1f I ￥jf ?  2 1f′ I ￥9f ?  3 I ￥ ?i ? 12 ,x x 9μ 21 12 () () ()( ) 1 f xfxfxxx′≥+ ? 13. ! 1f I  ￥=¨ V ? f ?5  I  1 j  ? f ?   b f ? () 0fx′′ > () 0fx′′ < xI∈ 14. ? f 0 x =¨ V? 5 ( 00 ,()x fx )1 wL y" f(x)￥.?￥A1Hq ^ b 0 ()0fx′′ = 15. ! ?f 0 x V?   #× ==¨ V? ? ?  ￥ ?|MQ5 ( 0 ()Ux 0 ()Ux + 0 ()Ux ? ()fx′′ 00 ,()x fx )1 wL y" f(x)￥.?b ? a '1 p  1. ' Y ?3 ?′? ?# s il D+? il g? ?? ? ?￥ ￡ ü ZE ?? ? ? -W￥c1"b 2. ? g ? L’Hospital E 5i ?? ?¨aá ?? ?1 p t?? T￥K 3. ' Y ?3 f ? B u W ? ?[ #?ì ?? ￥il ?H q ? g ?¨ ? ? ?  f ????D?? uW￥ZE ? ?¨f ?￥???￡ ü t?? Tb 4. ' Y ?3 Taylor ? ? g? Taylor  T ? ??]?[￥ Taylor  T# -W ￥μs 5. g?i ? :Btè¨?f ?? Taylor Z 7 Ti ?F[?¨b 6. ?¨ { Taylor ??[￥ Taylor  Té?í ?9 ?i9μ ?¨} Peanlo?[ ￥ Taylor  T p tf ?￥Kb 7. ? pf ? ￥′ DK ′ k bf ?′ ￥à Q | ¤ ′A1 Hq [#? Ba ? =  sHq g? pf ?′￥B?ZE??? ? 2¨?Ba ?= sHq ??f ?￥ ′DK′ ? ?¨f ?￥′ ??f ?￥K′ ? |¤′￥? ? sHq 9?¨ '￥ 3b 8. k bf ? j?￥ à Q g? f ? j?￥ +? ?N ?  ? p w L￥ .? ?? ¨ f ?￥j?￡ ü tμ1￥ 5b 1 a? ? è5  - 3 - "Q =SDyú ? [? ?Ds [?T Z? è 1. ￡ ü? ? T ln ,(0 ) ba b ba ab baa ?? << <<b s ü?? T V [? ? 1 ()ba b ? < lnb ? lna < 1 a (ba)? V n?[ ^f ? ln  uW [,  ′-μ 7 ^? uW￥é ? ^ V  [,  P¨ ?ì μ °?′? ?b x ]ab (ba? ) ln x ]ab ￡ ü  ! = 5 =()fx ln x '( )fx 1 x . ¨ ? ì μ °? ′ T ,μ[,]ab ln b a = ln =b ? lna 1 ξ ()ba? ()abξ< < ?y 111 baξ <<? ^μ ()ba b ? < lnb ? lna < ba a ? ' ()ba b ? < ln b a < ba a ? è 2 k￡ xxx π 2 sin >>  ? ? ? ? ? ? << 2 0 π x b s  ?  ?? T1 π 2sin 1 >> x x ? H0→x 1 sin → x x ? 2 π =x  x xsin -′ 1 π 2 ? ^1￡￥?? TM??1￡f ? () x x xf sin = -′o? 2 π D-W ￡ ü  I n f ? () ? ? ? ? ? ? ? = 1 sin x x xf  0 2 0 = << x x π  ? 2 0 π << x Hμ () ()0 cossincos 22 ' <?= ? = tgxx x x x xxx xf  ?[ ()xf ? ? ? ? ? ? 2 ,0 π =??h  ? ( )xf  ? ? ? ? ? ? 2 ,0 π  ?? ?[μ () () ? ? ? ? ? ? >> 2 0 π fxff ' π 2sin 1 >> x x  xxx π 2 sin >>  ' è9 V| ó ??? Ts1 Dxx sin> xx π 2 sin > ?￡ ü è 3 ￡ ü? ? T  a x ax? ≤ 1 a? (0,0 1xa><<) ￡ ü  ! 5() ( )aaxxxf a ???= 1 ( ) ( )1 11' ?=?= ?? aa xaaaxxf 7 ¤ ·() 0 ' =xf - 4 - "Q =SDyú ? [? ?Ds [?T Z? B? ?? H 1=x 10 << x ( ) 0 ' >xf  ? H1>x ( ) 0 ' <xf V 7 ^   ￥Kv′'μ ()1f ()xf (+∞,0 ) () ( ) 01 =≤ fxf ?[ a x ax? ?  1  ≤  a? 0 a x ax? ≤ 1 a? (0,0 1xa><<) è 4 ! ￡ ü?? T0,0 >> yx () 2 lnlnln yx yxyyxx + +≥+ O?|? yx = H? ?b s  | ? ? T H] H"[ 2M?11 ( ) 2 ln 22 lnln yxyxyyxx ++ ≥ + L V A PH ^f ?  ?  )￥′￥ ü ( ′7·H ^ ???() tttf ln= x y 2 yx+ )￥f ? ′ ? Ho3 ' V¤￡b () 0 " ≥tf ￡ ü  ! ,'() tttf ln= ( ) ttf ln1 ' +=  () 0 1 " >= t tf ,#? () ()[] ? ? ? ? ? ? + =+ 22 1 yx fyfxf V¤ () 2 ln 22 lnln yxyxyyxx ++ ≥ +  ' () 2 lnlnln yx yxyyxx + +≥+ ?|? yx = H? ?b è 5 pf ? ￥′b 2 2 x exy ? = ± ^ ? i=¨? ?￥f ? ?¨?= ?YE ?′?÷1ZLb 3   222 e)1)(1(2)2(ee2 2 xxx xxxxxxy ??? ?+?=?+=′ 7 3¤×??0=′y 1,0,1 ==?= xxx  2 e)251(2 42 x xxy ? +?=′′ Oμ 1 10 e4,02 ? ±== ?=′′>=′′ xx yy ?′￥?= ?YE? ^f ?￥l′?0=x 0 0 = =x y  1,1 =?= xx ^v′? v′ 1 1 ? ±= =ey x è 6 pf ? xyz = Hq 2=+ yx /￥′b - 5 - "Q =SDyú ? [? ?Ds [?T Z? 3E 1 ± ^  ?1íHq′ù5 p3 3  ? 2=+ yx V¤ xy ?= 2 } ? xyz = ¤ o3 p ￥′  7 2 2)2( xxxxz ?=?= 2 2 xxz ?= 022 =?=′ xz 3¤ ·B ×?? ,1=x ? 02 <?=′′z  a ü? f ?1=x xyz = Hq r?v′ Ov′?1 ?￥v′1 2=+ yx )1,1( 1=z 3E 2 ?¨ ?ì μ °e ?E p3 3  ! ?ì μ °f ? )2(),,( ?++= yxxyyxG λλ 3 ó ?Z?F ? ? ? ? ? =?+=′ =+=′ =+=′ 02 0 0 yxG xG yG y x λ λ λ 3¤·B×?? ?y1ù5' &iv′ ?[1,1 == yx 1,1 == yx ^f ? xyz =  Hq /￥′12=+ yx 1 )1,1( =z ?i  1f ?￥′Df ?￥Hq′ ^ ??]￥à Q B?f ? V[í′ ? B?￥Hq/ V[μHq′b  2 p3H q′ H B? V[¨ ?ì μ °e ?E? ^ 1?i ?¨ ?ì μ ° e ?E?é￡ùf ? ),(),(),,( yxFyxfyxG λλ += X? ^=íf ?  #? ? ?¨=íf ?′￥ sHq ? ?×?? ^?1′? N H V[? L=ù5é? ?Yb  3?= íf ?￥Hq′ù5 aa V[?1Bíf ? p′￥ù5 è ? è 6 ? ?1?? T¤￡ ü4 ? μr￥ZE P¨ H?è¨ '?ZE÷? a ? 4T \? ￥ís31? ?? T￥üa 8? T ?F[ê4μ￥ V[¨?ZE￡ üb ?aˉ 5 ￡ ü   Z ?  ^è ?  uW [0 =? V ?μ ??]￥ L ?  3 30xxc?+= c 0 ,1]  Z? n xpx q+ += 1?? ? 1 L ? ? 1 } ? Hà μ  ? Ln ,pq n - 6 - "Q =SDyú ? [? ?Ds [?T Z? ? ? 1  ? Hàμ ?? L?bn  ! () (1 ), , mn f xx xm=? n1?? ? [0,1]x∈ 5i (0,1)ξ ∈  P 1 m n ξ ξ = ?  ?¨ ?ì μ °?′? ?￡ ü/ ?? T   sin sin , , ( , );xyxyxy? ≤ ? ∈ ?∞ +∞    tan , ( , ), 22 xxx π π ≤∈??|? ?? O?? 0x =     1, 0 x exx>+ ≠;   ln ,0 ; yx y yx x y yxx ?? << <<   2 arctan , 0. 1 x xxx x << + >  !f ? ? μ ??￥=¨? ?￡ üa '' 2 0 ()()2() lim ( ). h fa h fa h fa f a h → ++ ?? =   ! ' lim ( ) x f xa →+∞ =  p￡ ?i μ0T > lim [ ( ) ( )] . x f xT fx Ta →+∞ +? = f ?  [, V? ? ￡ üi()fx ]ab 0a ≥ (,)abξ ∈  P¤ 22' 2[ () ()] ( ) ().fb fa b a fξ ξ?=?   !  (,  V? O()fx )a +∞ lim ( ) lim ( ) xxa f xfx + →∞→ A= = b p￡ i (, )aξ ∈+∞ P b ' () 0f ξ =  ! V? p￡   ,?-WB?μ()fx ()fx ' () ()f xfx+ ￥ ,?  !f ? ()fx 0 x ?í ?? " 0 x ?? V?  O 0 ' lim ( ) xx f x → A=  p￡ ' 0 ()f x i  O ' 0 ()f xA=   ?  [, V? O()fx ]ab '' () ()f afb≠  1o?k ' ()f a ? ' ()f b -W￥ ?B L ?5à iB? (,)abξ ∈  P ' ()f kξ =  - 7 - "Q =SDyú ? [? ?Ds [?T Z?  !f ?  (, = V? O()fx )ab ' ()f x ??￡ ü ' ()f x  (, ??)ab  ? f ?  ?  [, ??  V? ￡ üi()fx ()gx ()hx ]ab (,)ab (,)abξ ∈  P¤ ''' () () () () () () 0 () () () fa ga ha fb gb hb fghξξξ =   !  (, ??  O()fx )?∞ +∞ lim ( ) x fx →±∞ = +∞￡ ü  (,  | ? ?￥Kl′ ()fx )?∞ +∞  !  [, ??()fx )ab lim ( ) xb f xB ? → =    ?i 1 [,)x ab∈  P 1 ()f x> B5  [, r?Kv′ ()fx )ab   ?Ti  1 [,)x ab∈  P 1 ()f xB=  ??y  [, r?Kv′$()fx )ab  !  [, μ?()fx )a +∞ ' ()f x i O ' lim ( ) x f x →+∞ b=  p￡ 0b =  p￡ arcsin arccos ( 1) 2 xxx π +≡≤  p/  ???￥K   0 tan lim ; sin x ax bx →    2 3 0 1cos lim ; sin x x x x → ?    0 ln(1 ) lim ; cos 1 x x x x → + ? ?    0 tan lim ; sin x x x x x → ? ?    0 11 lim ; 1 x x x e → ?? ? ?? ? ??    0 ln cos lim ; ln cos x ax bx →    2 tan 6 lim ; sec 5 x x x π → ? +    1 11 lim ; ln 1 x xx → ?? ? ?? ? ??  - 8 - "Q =SDyú ? [? ?Ds [?T Z?   lim( ) tan ; 2 x x x π π → ?    1 1 1 lim ; x x x ? →  f ?  [0()fx , ]x ?¨ ?ì μ °?′? ?μ ' () (0) ( ), (0,1).fx f f xxθθ?= ∈ k￡/ f ?μ 0 1 lim 2 x θ → =    () ln(1 );fx x=+   () . x f xe=   ! =¨ V? p￡()fx '' 2 0 (2)2( ) () lim ( ). h fx h fx h fx f x h → +? ++ =  / f ?? ?¨ ?ArE5 pK   2 0 1 sin 2 lim ; sin x x x →    sin lim ; cos x x x x x →∞ + ?    sin 2sin2 lim ; (2 sin ) x x xx xxe →∞ + +    2 1 (1)sin lim . ln 1 sin 2 x x x x π → ? ?? + ?? ??  - 9 -