5 í ??f ?   ??/ f ?￥1 ??l×   22 1 )ln( yx x xyu ?? +?=    zyx u 111 ++=     )( 22222222 rRrzyxzyxRu >?+++???=     22 arcsin yx z u + = b 3 (1) { }xyyxyxD ><+= ,1),( 22 b (2) {}0,0,0),,( >>>= zyxzyxD b (3) { } 22222 ),,( RzyxrzyxD ≤++≤= b (4) { }0,),,( 2222 ≠++≤= yxyxzzyxD b  ! 2/322 3 )( yx x x y f + = ? ? ? ? ? ?   p b)0( >x )(xf 3 y1 3 3223/2 2 2 1 () 1 yx f xxy y x ?? == ?? + ?? ? ? ?? + ? ? ?? ?? ? ? ? ?  ?[ 2 3 2 )1( 1 )( x xf + = b  ?f ? )1(),( ?+= xfyyxz  O? H  p ? b4=y 1+= xz )(xf ),( yxz 3 ? (,4) 4 ( 1) 1z xfx=+ ?=+x V¤ 2 (1) 1(11)fx x x?=?= ?+ ?1 ?[ 22 () ( 1) 1 2f xx x=+?=+x, 1),( ?+= yxyxz b  ) ?/ f ?? t? H￥K ^?i),( yx )0,0( 1   yx yx yxf + ? =),(    22 ),( yx xy yxf + =        ? ? ? << = ;0 ,0,1 ),( 2  ?? xy yxf 84 33 ),( yx yx yxf + = b 3  1? ? 1 (, ) 1 x kx k fxkx x kx k ?? == ++ G ?? L ? [ ? t? Hf ?K?ib ),( yx )0,0(  2 2 22 (, ) () 1 kx k fxkx 2 x kx k == ++ G ?? L ?[? t? Hf ? K?ib ),( yx )0,0(  3?? 2 (, ) 1 2 x fx =  ?[?  wL),( yx 2 2 x y = t? Hf ? K1 7?  x à t? Hf ?K1  ?[? t ? Hf ?K?ib )0,0( ),( yx )0,0( ),( yx )0,0(  4 ?¨ ü (′?? T = + 3 84 yx 3 88 844 4 1 3 2 1 2 1 yx yxx ≥ ++  V¤ 133 3 33 3 848 3 ||4||4 |(,)| | | 33 || xy xy f xy xy xy xy =≤ = + 0,(( , ) (0,0))xy→→ ?[? t? Hf ?Ki O1 b ),( yx )0,0(  íf ?￡ üK·B?  ?μ?? ? ??? ? C/?b ￡  L ! f (x)=A  f (x)=#5 0 lim xx→ 0 lim xx→ 0ε? >  101 : |() |fAε? <x  0, (0 | | )δ δ?>? <? <xxx 202 : |() |fBε? <x b 0, (0 | | )δ δ?>? <? <xxx | { } 12 min , 0δδδ=>? 0 0| |δ<? <xx ? ? | ||() ||() |2AB f A f B ε? ≤?+?<xx ?? ε1 ?i? ? ?[ A=B'K·Bb  2L ! f (x)=A 5? 0 lim xx→ 1ε =  2 0 0, (0 | | ) :δ δ?>? <? <xxx |() |1fA? <x  ' |()|| |1fA< +x b ?[ f (x) x ?￥ ? ?? 5×μ?b 0 (3) ! f (x)=A> g (x)=B5? 0 lim xx→ 0 lim xx→ 0 2 AB ε ? = >  101 : |() |fAε? <x  0, (0 | | )δ δ?>? <? <xxx ' () 2 A B fAε + >?=x b ? 202 : |() |gBε? <x  0, (0 | | )δ δ?>? <? <xxx ' () 2 A B gBε + <+=x | { } 12 min , 0δδδ=>? 0 0| |δ<? <xx ? ? ? ?? () () 2 AB gf + <<x x b (4)L?i 0ρ >  P? 0 0| |ρ<? <xx H? ? () () ()gfh≤≤xxx O g (x) = h (x)=Ab 0 lim xx→ 0 lim xx→ 0ε?>, ? h (x)=A 0 lim xx→ 101 : |() |hAε? <x  0, (0 | | )δ δ?>? <? <xxx ?[ ()hAε<+x b ?? g (x) =A 0 lim xx→ 202 : |() |gAε? <x  0, (0 | | )δ δ?>? <? <xxx ?[ ()gAε>?x b | { } 12 min , , 0δρδ=>? 0 0| |δ< ?<xx ? ? () () ()Ag fhAε ε?< ≤ ≤ <+xxx  ' f (x)=Ab 0 lim xx→  íf ?￡ üK￥ 15 ?E5L !? x t? x Hf ? f (x)? g (x)￥Ki5 0   f (x)g (x)) = f (x) g (x)  0 lim xx→ 0 lim xx→ 0 lim xx→ 3   (f (x) ·g (x)) = f (x)· g (x) 0 lim xx→ 0 lim xx→ 0 lim xx→   (f (x)g (x)) = f (x) g (x)  g (x) ? 0b 0 lim xx→ 0 lim xx→ 0 lim xx→ 0 lim xx→ ￡ L ! f (x)=A g (x)=Bb5 ?i 0 lim xx→ 0 lim xx→ 0ε >  101 : |() |fAε? <x  0, (0 | | )δ δ?>? <? <xxx 202 : |() |gBε? <x  0, (0 | | )δ δ?>? <? <xxx | { } 12 min , 0δδδ=>? 0 0| |δ<? <xx ? ? |(() ()) ( )|| () | | () |2fg ABfAgBε±?±≤?+?<xx x x  ?[ 1? ?b ?? g (x) x μK ?[ g (x) x 0 ?μ?'i? ? X ? 0 '0δ > ?x 0 (0 | | ') :δ<? <xx |()|gX<x b | { } 12 min ', , 0δδδ= > ? 0 0| |δ<? <xx ? ? | ()() || ()() ()| | () |fg ABfg Ag Ag AB?≤ ? + ?xx xx x x (||)X A ε<+  ?[ 2? ?b ?? B?0 0 2 B εε ?? ?<< ?? ??  "0δ? >  0 (0 | | ")δ? <? <xxx  || |()|| | 2 B gBε>?≥x b | { } 12 min ", , 0δδδ=>? 0 0| |δ< ?<xx ? ? () (() ) (() ) () () f ABf AAg B gB Bg ?? ? ?≤ xxx 2 2(||||) || AB B ε + <  ?[ 3? ?b  p/ òK   22 )1,0(),( 1 lim yx xy yx + ? →    22 22 )0,0(),( 1 lim yx yx yx + ++ →     xy xy yx 11 lim )0,0(),( ?+ →      11 lim 22 22 )0,0(),( ?++ + → yx yx yx     22 2 )0,0(),( )ln( lim 2 yx ex y yx + + →    22 33 )0,0(),( )sin( lim yx yx yx + + →     2222 22 )0,0(),( )( )cos(1 lim yxyx yx yx + +? →     b )(22 )(lim yx y x eyx +? +∞→ +∞→ + 3  1 (,) (0,1) 22 22 (,) (0,1) (,) (0,1) lim (1 ) 1 lim 1 lim ( ) xy xy xy xy xy xy xy → → → ? ? == ++ b 4  2 22 22 (,) (0,0) (,) (0,0) lim ( ) 0, lim (1 ) 1 xy xy xy xy →→ + =+= ?[ 22 22 )0,0(),( 1 lim yx yx yx + ++ →  ∞+ b  3 (,)(0,0) (,)(0,0) 11 1 lim lim 11 xy xy xy xy xy →→ +? = + +  2 1 b  4 22 22 22(,)(0,0) (,)(0,0) lim lim ( 1 1) 2 11 xy xy xy xy xy →→ + =+++ ++? = 2 b  5 22 222 ln( ) ln(1 1) ( ) ( ) yy x exexyoyxyox+=++?=++ =+++y ?[ 2 2 22 (,) (0,0) ln( ) lim y xy xe xy → + = + 1b  6 33 33 22 22 |sin( )| | | | || | 2| || |x yxyxyxyxy xyxy+≤+=++?≤++ ?[ 22 33 )0,0(),( )sin( lim yx yx yx + + →  b 0  7y1 () 22 222 1 1cos()()(,)(0, 2 xy xy xy?+ + →～ ) 222 2222 1 () 1 2 ()|| xy x yxy xy + ≥ + (,) (0,0) 1 lim xy xy → =+∞  ?[ 22 2222 (,) (0,0) 1cos( ) lim () xy xy xyxy → ?+ = + 222 2222 (,) (0,0) 1 () 2 lim () xy xy x yxy → + = + ∞+ b  8 22() 2 2 lim ( ) lim ( ) lim ( ) 0 xy x y y x xxx yy xye xee yee ?+ ? ? ? ? →+∞ →+∞ →+∞ →+∞ →+∞ →+∞ ????+= + ???? =b  ) ?/ f ?e?￥=×K?=QK 5   222 22 )( ),( yxyx yx yxf ?+ =     22 2222 )1()1( ),( yx yyxx yxf + +?+ =     x y y xyxf 1 sin 1 sin),( += b 3 (1) ?? 2 42 4224 00 (1 ) 1 lim ( , ) lim (1 ) 1 xx yxkx xkx fxy 2 x kx k x k →→ =+ + == + ++  ?[=×K?ib ? 2 0 0 lim ( , ) 0, 0 x fxy y y → == ≠ V? 00 lim lim ( , ) 0 yx fxy →→ = b] ? V?  b ?[=QKi O??? b 00 lim lim ( , ) 0 xy fxy →→ =  2?? 2 2 22 22 2 22 00 (1 ) (1 ) 1 lim ( , ) lim (1 ) 1 xx ykx 2 x xkx kx k fxy x kk →→ = +? + ? == + +  ?[=×K?ib?   2 00 0 lim lim ( , ) lim(1 ) 1 yx y fxy y →→ → =? + =?  b 2 00 0 limlim ( , ) lim(1 ) 1 xy x fxy x →→ → =+= ?[=QK?i??M?b  3?? |  ?[(,)|||||fxy x y≤+ 0 0 lim ( , ) 0 x y fxy → → = b ?? 0 1 lim sin ( 0) x yy x → ≠ ? 0 1 lim sin ( 0) y xx y → ≠ ??i ?[ ?=Q K??ib ￡f ? 6 ? ? ? ? ? ? ? ? ? <<>? ≤<> ? ? ? ? ? ? ? =  ?? O O ,0 ,20),2( 1 , 2 1 0, 2 12 ),( 222 2 222 2 xyxxyx x xyxxxy x yxf  e?? ??7  ?? ??b ￡  ! 0x > 2 (, )fxx = 22 2 21 1 2 xx x ?? ? = ?? ??  ?[?? (, )x y  t ?e? Hf ? ￥K1 7?? 2 (0yxx=>) (, )fxy (, )x y  à t?e? Hf ? ￥K1  ?[f ? e?? ??b x (, )fxy (, )fxy ?f ?   ??￥ ???o1 I nf ?/ ? wL(, )fxy 22 2 (0x > 1 ,,2 2 yxyxyx=== ) ￥ f ? y1" ?  ? wL?e ?￥ u× f ?A ? ?? b ! b 0 0x > 2 00 0 0 1 (, )(, ) 2 x yxx= ??? 00 00 2 2 00 2 (,) ( , ) (,) ( , ) /2 1 2( ) 2 lim ( , ) lim 0 ( , ) xy x y xy x y yx yx f xy f x y x →→ > ? === 00 2 (,) ( , ) /2 lim ( , ) 0 xy x y yx fxy → ≤ = 00 (, )f xy=  ?[f ? (, )fxy 00 (, )xy= 2 00 1 (, ) 2 x x ??b] ? V?f ? (, )fxy 00 (, )xy= 2 00 (,2)x x 9 ??b  00 (, )xy= 2 00 (, )x x ??? 00 00 22 222 00 22 (,)(,) (,)(,) 0 2 22 lim ( , ) lim 1 xy xy xy xy xyx xxxy fxy xx →→ >> ?? === 00 (, )f xy=  00 00 22 222 00 22 (,)(,) (,)(,) 0 /2 11 2( ) 2( ) lim ( , ) lim 1 xy x y xy x y xyx yx x x fxy xx →→ <≤ ?? === 00 (, )f xy=  ?[f ? (, )fxy 00 (, )xy= 2 00 (, )x x 9 ??b 8  ? ?f ? " e?? ??  ??? ??b(, )fxy ) ? f ? ? ? ? ? ? =+ ≠+ + = 0,0 ,0, ),( 22 22 22 2 yx yx yx yx yxf  ￥ ??S?b 7 3 A ?f ?  u×(, )fxy { } 22 (, ) 0xy x y+ ≠  ?? ?[o1 I nf ? e?￥ ???b?(, )fxy 22 1 ||||( 2 2 )x yxxy≤+¤? 2 22 1 || 2 xy x xy ≤ +  ?[ (,) (0,0) lim ( , ) xy f xy → = 2 22 (,) (0,0) lim 0 xy xy xy → = +  'f ?e?9 ??byNf ?  ü ? ?? ??b(, )fxy  !  uW  μ ??? ?)(tf ),( ba ),(),( baba ×=D b?l ￥ f ? D ? ? ? ? ? =′ ≠ ? ? = .),( ,, )()( ),( yxxf yx yx yfxf yxF   ￡ ü? ?? ? ?),( bac∈ )(),(lim ),(),( cfyxF ccyx ′= → b ￡ ?5 ! ?¨ Lagrange?′? ? () () '()( )fx fy f x yξ? =? ? ξo ? ? -Wb ?[ x y (,) (,) (,) (,) lim ( , ) lim '( ) ( ) xy cc xy cc xy Fxy f f cξ →→ ≠ ′= =  (,) (,) lim ( , ) lim '( ) ( ) xy cc x c xy Fxy f x f c →→ = ′= =  8?  ?  T V¤ )(),(lim ),(),( cfyxF ccyx ′= → b   !=íf ?  7"),( yxf 2 RD ? =?M  ^ ??￥?M  x y ?@ LipschitzHq |),(),(| yxfyxf ′′?′ ? |'| yyL ′′?  ? D∈′′′ ),(),,( yxyx L1è ? Yè?1 Lipschitz è ? b￡ ü  = ??b),( yxf D ￡ L ! ??f ??M  ^ ?? 00 (, )xy∈D x 0ε?>  0 0, (| | )xx xδ δ?>? ? < ? ? 000 (, ) ( , )f xy f x y? ε< b 8 ? 00 (, ) ( , ) min(, )xy x y δ ε?< H ≤? ),(),( 00 yxfyxf +? ),(),( 0 yxfyxf 00 (, ) ( , ) 0 f xy f x y? 0 Ly y≤?+ 00 (, ) ( , ) 0 f xy f x y? Lε ε≤+ ?[ ),( yxf 00 (, )x y = ??￡8b 13￡ ü ? f? g ^ D ￥ ??? 5?  f + gDf ? ! f , g# D ? ^ ??￥b ￡ L ! D? f? g ^ ?? 0 x ∈ 0ε? >  0 0, (| | )δ δ?>? ? <xx x ? ? 0 |() ()|ε? <fx fx  1 0, (| | ) 01 δ δ?>? ? <xx x ? ? 0 |() ( )|ε? <gx gx  ? ^ 00 |() ()(() ()|+ ?+fx gx fx gx  00 | ( ) ( )| | ( ) ( )|≤? +?f x f x g x g x 2ε≤  ?[?  f + g ??b? 0 x 00 |(),() (),()<>?<fx gx fx gx |> 0 |   00 |() (),() (),()()=< ? >+< ? >fx fx gx fx gx gx 0 |()| | ( )|ε ε≤+gx f x  ?? g ?? ? [ g￥ ??s ? ?? V7? ?μ? ? ^ g9 ?μ?b?  T! f , g# ??￡8b 0 x ￡ ü ˉ?? ￥ ???? ? ? ? b ￡ L ! g D  ?? f ?  ?? i O 000 ,()D∈ =∈xugx?b? f   ?? 0 u 0 0, 0, (| | )ε ηη? >?>? ? <uu u ? ? 9 0 |() ()|ε? <fu fu b ?  ? 0η > ? g ??? 0 x 0 0, (| | )δ δ? >? ? <xx x ? ? 0 |() ( )|η? <gx gx b ? ^? 0 ||δ?<xx H 00 | ( ) ( )| | ( ) ( )| ε?=?nullnullf gx f gx fu fu < ?[ˉ?f ? nullf g ??b 0 x 10