5 í kl Dí kv ￥¨ 1. ?? aD α P/ òí kl í kv ?N? (j) a x α  (1) u(x) = x x x 54 32?+ 3 , (x10 x1?) (2) u(x) = xx xx 52 4 2 3 + ? 3 (x10 x1?) (3) u(x) = x 3 + x 23 (x10+ x1+?) (4) u(x) = xxx++(x10+ x1+?) (5) u(x) = 13+ x - 12 3 + x (x10 x1+?) (6) u(x) = x 2 1+ - x (x1+?) (7) u(x) = 3 x x+ - 3 2 x (x10+) (8) u(x) = 1+ xx - e (x10+) 2x (9) u(x) = ln cos x - arc tan x 2 (x10) (10) u(x) = xtan1+ - 1?sin x (x10)b 3  1 j  j b )(xu )0(2 3 →xx )(xu )( 5 ∞→xx  2 j  j)(xu )0(2 1 →? ? xx )(xu )( 3 1 ∞→xx b  3 j)(xu )0( 3 2 +→xx  j)(xu )( 2 3 +∞→xx b  4 j)(xu )0( 8 1 +→xx  j)(xu )( 2 1 +∞→xx b  5 j)(xu )0( 6 5 →xx  j)(xu )(3 2 1 +∞→xx b  6 j)(xu )( 2 1 1 +∞→ ? xx b  7 j)(xu )0( 2 1 +→xx b  8 j)(xu )0(2 +→? xx b 47  9 j)(xu )0( 2 3 2 →? xx b  10 j b )(xu )0( →xx 2. (1) ? x1+? H / M ? ^í kv  | ? ìV?¨?ú¨é ? ? i a ü ??b a x (a# 1), x x , x α (α# 0) ln k x (k# 0) [x]! (2) ? x10+ H / M ? ^í kl  | ? ìVú¨??¨é ? ? i a ü ??b x α ( # 0)α ! 1 1 ? ? ? ? ? ? x , a x ? 1 (a# 1) x x 1 1 ? ? ? ? ? ? ? , ? ? ? ? ? ? ? x k 1 ln (k# 0)b 3  1? x1+? HV?¨í kv ?ú¨í kv ￥ ? 1 ln k x (k# 0) x α (α# 0) (a# 1), [x]!, a x x x b ￡ ü ! 1+<≤ nxn 5 nx a n a x αα )1( 0 + <<  !]![ 0 1 n a x a nx + <<  nx n n x x )!1(]![ 0 + << b ? ∞→n lim 0 )1( = + n a n α  ∞→n lim 0 ! 1 = + n a n D ∞→n lim 0 )!1( = + n n n '¤? x x a x α +∞→ lim 0=  ∞→n lim 0 ]![ = x a x  ∞→n lim 0 ]![ = x x x ] H9¤? α x x k x ln lim +∞→ 0 )( lim == +∞→ y k y e y α b )ln( xy =  2? x10+ HVú¨í kl ??¨í kl ￥ ? 1 x x 1 1 ? ? ? ? ? ? ? , ! 1 1 ? ? ? ? ? ? x , a x ? 1 (a# 1) x α (α# 0) ? ? ? ? ? ? ? x k 1 ln (k# 0)b ￡ ü 7 x y 1 = 5? x10+ Hμ +∞→y b? I 1￥ ? ' V¤ ? 2￥ ? b 3. 9 ?/ K 48 ò lim x→0 112 13 23 +? + + xx xln( )  ó lim x→0 1 1 ? ? cos cos x x  ? lim x→+∞ ( xxx++- x ) ? lim x→+∞ ( 1 2 ++xx- 1 2 ?+xx) ? lim x→α aa x x ? ? α α (a# 0) × lim xa→ xa xa αα ? ? (a# 0) ? lim x→+∞ x ( ln (1+x) - ln x ) ù lim xa→ ln lnx xa a? ? (a# 0) ú lim x→0 (e)x x x + 1  ? lim x→0 2 1 2 2 cos x x x ? ? ? ? ? ? ? ? ?  ü lim n→∞ n ( x n - 1) (x# 0) Y lim n→∞ n 2 ( x n - x n+1 ) (x# 0)b 3  1 lim x→0 = + +?+ )31ln( 211 3 2 x xx lim x→0 )31ln( )121()11( 3 2 x xx + ?+??+ 0 lim → = x = ? x xx 3 3 2 2 1 2 6 1 b  2 lim x→0 = ? ? x x cos1 cos1 lim x→0 = +? ? )cos1)(cos1( cos1 xx x lim x→0 = + )cos1( 2 1 2 1 2 xx x 0b  3 (lim x→+∞ xxx++- x ) +∞→ = x lim = +++ + xxxx xx lim x→+∞ = x x 2 2 1 b  4 (lim x→+∞ 1 2 ++xx- 2 1 xx +? ) +∞→ = x lim = +?+++ 22 11 2 xxxx x 1b  5 α→x lim = ? ? α α x aa x α→x lim = ? ? ? α αα x aa x )1( α→x lim ()lnax a x α α α ? = ? aa ln α b  6 lim xa→ = ? ? ax ax αα lim xa→ = ? ? ax ea a x )1( lnα α lim xa→ ax a ax a ? ? + )1ln(α α ax→ = lim = ? ? ? ax a ax a α α 1?α αa b  7 x ( ln (1+x) - ln x )lim x→+∞ = + = +∞→ x x x 1 ) 1 1ln( lim 1b 49  8 lim xa→ ln lnxa xa ? ? lim xa→ = ? ? + ax a ax )1ln( a 1 b  9 lim x→0 =+ x x x 1 )e( lim x→0 =?++ x x x 1 )1e1( lim x→0 =+ x x 1 )21( 2 e b  10 lim x→0 = ? ? ? ? ? ? ? ? ? 2 1 2 2 cos x x x lim x→0 2 1 2 2 )cos1(1 x x x ? ? ? ? ? ? ? ? ??? 0 lim → = x ()=? 2 1 2 1 x x 1? e b  11 n (lim n→∞ x n - 1) )1(lim ln 1 ?= ∞→ x n n en =?= ∞→ )ln 1 (lim x n n n xln b  12 (lim n→∞ n 2 x n - x n+1 ) ? ? ? ? ? ? ? ? ???= + ∞→ )1()1(lim ln 1 1 ln 1 2 x n x n n een = ? ? ? ? ? ? ? ? ? ? ? ? + ?= ∞→ 1 11 lnlim 2 nn xn n xln b 50 5 > uW ￥ ??f ? 1. ￡ ü !f ? fx() ),[ +∞a  ?? O = A μK ? 5  μ?b lim x→+∞ fx() fx() ),[ +∞a ￡ ? = A μK ?  V ?lim x→+∞ fx() aX >?  Xx >?  1)( <? Axf ' 1)(1 +<<? AxfA b? > uW ￥ ??? V?  μ?'  )(xf ],[ Xa )(xf ],[ Xa ],[ Xax∈? Bxf <)( b 7 }1,max{ += ABM  5}1,min{ ??= ABm [ )+∞∈? ,ax ? ? Mxfm << )( b 2. ￡ ü ?f ?  7 uW  ?? O f(a+)? f(b-)i5 ? V |?o? f(a+)? f(b-)-W￥B M?W′b fx() ),( ba ￡ 7 ? ? ? ? ? =? =+ ∈ = bxbf axaf baxxf xf )( )( ),()( )( ~  5 )( ~ xf > uW ???^ !],[ ba )()( ?<+ bfaf ?> uW  ??f ?￥?W′? ? V? )( ~ xf > uW  V |? ￥ B M′? ^  7 uW  V |?o? f(a+)? f(b-)-W￥B M?W′b ],[ ba )](),([ ?+ bfaf fx() ),( ba 3. ￡ ü ?> uW ￥??μ?f ? ? |? f(a)? f(b)-W ￥B M′5 ^ ￥ ??f ?b ],[ ba fx() fx() ],[ ba ￡ ?¨Q￡Eb ?^ ! ??9Fb ?)(xf ),( ba∈ξ ^ ￥? ??? 5 )(xf )( ?ξf D )( +ξf ?i O )()()()( bfffaf ≤+<?≤ ξξ ? ^ |? ? 7 uW )(xf ))(),(( +? ξξ ff ?s? )(ξf ￥′ DHq ± ? ^ ￥ax = )(xf 51 ? ??? 5 i O)( +af )()()( bfafaf ≤+< ? ^ |?? 7 u W ?￥′ 9DHq ± ]" V[￡ ü )(xf ))(),(( +afaf bx = 9? V ? ^ ￥? ???b )(xf 4. ?¨ Bolzano-Weierstrass ? ?￡ ü> uW  ??f ?￥μ??? ?b ￡ ?¨Q￡Eb ! > uW  ???í?5i?    ?@ )(xf ],[ ba {} n x ],[ bax n ∈ nxf n >)( ' ∞= ∞→ )(lim n n xf b? Bolzano-Weierstrass ? ?i0  { } k n x  ξ= ∞→ k n k xlim  O ],[ ba∈ξ by1 ?)(xf ξ ?? ?[μ )()(lim ξfxf k n k = ∞→ D ∞= ∞→ )(lim n n xf á 3 ±b 5. ?¨> uW*? ?￡ ü ,?i? ?b ￡ ! > uW  ?? O)(xf ],[ ba 0)()( <bfaf ? ^ !    b 1 aa = 1 bb = 0)( <af 0)( >bf ?T 0) 2 ( 11 = +ba f 5? ?¤￡b ?T 0) 2 ( 11 < +ba f 5 7 2 11 2 ba a + =   ?T 12 bb = 0) 2 ( 11 > +ba f 5 7 12 aa =  2 11 2 ba b + = b ?T 0) 2 ( 22 = +ba f  5? ?¤￡b ? T 0) 2 ( 22 < +ba f 5 7 2 22 3 ba a + =   ?T 23 bb = 0) 2 ( 22 > +ba f 5 7 23 aa =  2 22 3 ba b + = b ,"" ?"￥V? V[B°é?/ ?b ?Ti ?  P ¤k 0) 2 ( = + kk ba f  5? ?¤￡ ?T?i ?  P¤k 0) 2 ( = + kk ba f 5¤?B?> uW* {  ?@}],[ nn ba 0)( < n af  b?> uW*? ? V?i0)( > n bf 52 ·B ?? ?μ> uW ￥?],[ nn ba ξ O = ∞→ n n alim ξ= ∞→ n n blim b?  ? )(xf ξ￥ ??? V? =)(ξf 0)(lim ≤ ∞→ n n af D =)(ξf 0)(lim ≥ ∞→ n n bf V7¤? 0)( =ξf ? ?¤￡b 6. ￡ üZ?  à μB???b bxax += sin 0, >ba ￡ 7 bxaxxf ??= sin)( 5 )(xf ),0[ +∞  ??b | 5  ? ,?i? ?  à μB??b baA +> 0)0( <f 0)( >Af )(xf ),0( A 7￡ üZ?  μ O?μB? L?b 0 3 =++ qpxx 0>p ￡ 7 5 qpxxxf ++= 3 )( )(xf ),( +∞?∞  ^?ì??9F￥b? ?∞= ?∞→ )(lim xf x +∞= +∞→ )(lim xf x ^? )(xf ),( +∞?∞ μ O?μB? L?b 8￡ ü  1 sin 1 x  0,1 ?Bá ??? (a,1)(a# 0) Bá ??  2 sin x 2  ?Bá ??? [0,A] Bá ?? ),( +∞?∞  3 x [ Bá ?? )0,+∞  4 ln x [ )+∞,1 Bá ?? (5) xcos  [ Bá ??b )0,+∞ ￡  1  7)1,0( πn x n 1 ' =  2 1 " π π + = n x n  ? 0 "' →? nn xx 1 1 sin 1 sin "' =? nn xx  ?[ sin 1 x  0,1 ?Bá ??b  (a# 0)  )1,(a 0>?ε  | 0 2 >= εδ a )1,(, 21 axx ∈?  δ<? 21 xx  ? ? 2 21 2121 111 sin 1 sin a xx xxxx ? ≤?≤? ε<  53 ?[ sin 1 x  (a,1) (a# 0) Bá ??b  2  7),+∞∞? 2 ' π π += nx n  πnx n = " 5 ? 0 "' →? nn xx () ( ) 1sinsin 2 " 2 ' =? nn xx  ?[ sin x 2  ?Bá ??b ),( +∞?∞  ],0[ A 0>?ε  | 0 2 >= A ε δ  ],0[, 21 Axx ∈?  δ<? 21 xx ? ? 21 2 2 2 1 2 2 2 1 2sinsin xxAxxxx ?≤?≤? ε<  ?[ sin x 2  [0,A] Bá ??b  3  0>?ε  | 0 2 >=εδ [ )+∞∈? ,0, 21 xx  δ<? 21 xx ? ? 2121 xxxx ?≤? ε<  ?[ x [ Bá ??b )0,+∞  4  0>?ε  | 0>=εδ  [ )+∞∈? ,1, 21 xx  δ<?≤ 21 0 xx ? ? 21 2 21 21 1lnlnln xx x xx xx ?≤ ? ? ? ? ? ? ? ? ? +=? ε<  ?[ ln x [ Bá ??b )+∞,1  5  0>?ε  | 0 2 >=εδ [ )+∞∈? ,0, 21 xx  δ<? 21 xx ? ? 212121 coscos xxxxxx ?≤?≤? ε<  ?[ xcos  [ )0,+∞ Bá ??b 9￡ ü?? =￥ ?iB? Pi??V P ￥BH? P¤ P ^??￥??b ￡ V P?T? !?D à￥C?1x θ P?|?s?é1 )( 1 θl ? )( 2 θl ￥ L  5 )()()( 21 θθθ llf ?=  [ ]π,0 ?? ?@ )()0( πff ?= ? 54 ^Aμ ∈ 0 θ []π,0  ?@ 0)( 0 =θf 9ü ^ )( 01 θl )( 02 θl= b 10 !f ?  [0,2]  ?? O f(0) = f(2)￡ üi   P¤ b fx() ]2,0[, ∈yx 1=? xy )()( yfxf = ￡ 7 )()1()( xfxfxF ?+= 5 )(xF [ ]1,0  ?? )0()1( FF ?= ? ^A μ  ?@ b 7]1,0[ 0 ∈x 0)( 0 =xF 1 00 += xy 5 ]2,0[, 00 ∈yx   P¤ b 1 00 =? xy )()( 00 yfxf = 11 ?f ? μK 7 uW Bá ?? 5  μ?b fx() ),( ba fx() ),( ba ￡ ?  Bá ?? V?fx() ),( ba )( +af  )( ?bf i OμKb 7 ? ? ? ? ? =? =+ ∈ = bxbf axaf baxxf xf )( )( ),()( )( ~  5 )( ~ xf > uW ?? ?[],[ ba )( ~ xf  μ?yN   μ?b ],[ ba fx() ),( ba 12￡ ü  1  uW  ?Bá ??f ?-?A?Bá ??  2  uW  ?Bá ??f ?-?B?Bá ??b ￡  1 !f ?   uW)(xf )(xg I Bá ??5 0>?ε  0>?δ  Ixx ∈? ",'  δ<? "' xx ? ? 2 )"()'( ε <? xfxf  2 )"()'( ε <? xgxg ? ^ ε<+?+ )]"()"([)]'()'([ xgxfxgxf  ?[  uW)()( xgxf + I Bá ??b  2 !  uWxxgxf == )()( [ )+∞= ,0I 5   uW)(xf )(xg I Bá ???  uW 2 )()( xxgxf = I ?Bá ??b 13. !f ?   ?? Ofx() ],[ ba ],[,0)( baxxf ∈≠ ￡ ü  fx() ],[ ba 55 ???μb ￡ !  ? ??|5ifx() ],[ ba ],[",' baxx ∈  ?^ !  "' xx < P D ?]|?> uW  ??f ?￥?W′? ?A?i )'(xf )"(xf  ]",'[ xx∈ξ  P ¤ 0)( =ξf ?üá 3 ± ?[  A?? fx() ],[ ba ??μb 14 !f ?   ??fx() ],[ ba bxxxa n ≤<<<≤" 21 ￡ ü ? Aμ ],[ ba ξ P¤ )]()()([ 1 )( 21 n xfxfxf n f +++="ξ b ￡ ? > uW  ??f ?￥?W′? ? > uW  ??f ?B? ? |?Kv′?Kl′-W ??B?′b?? )}({min ],[ xf bax∈ []≤+++≤ )()()( 1 21 n xfxfxf n ")}({max ],[ xf bax∈  ?[ ?Aμ],[ ba ξ P¤ )]()()([ 1 )( 21 n xfxfxf n f +++="ξ b 15 ? f ?   ?? O = A(μK ? )5  Bá ??b fx() ),[ +∞a lim x→+∞ fx() fx() ),[ +∞a ￡ ? Axf x = +∞→ )(lim  :",',,0 XxxaX >?>?>?ε ε<? )"()'( xfxf b? ? )(xf  ?? ?[Bá ??9ü ^ [1, +Xa ] [ ] :)"'(1,",',10 δδ <?+∈?<<? xxXaxx ε<? )"()'( xfxf b? ^ [ ) :)"'(,",' δ<?+∞∈? xxaxx ε<? )"()'( xfxf b 56