"Q =SDyú ? [? ?Ds [?T Z? ? ?cf ?K B a 'à Q  1. f ?K ￥?l òx→+∞ H f ?K ￥ ? l ! 1?l  [,f )a +∞ ￥f ? &1 L ?b ? ? ó￥ 0ε > i? ?2  P¤? Hμ (a≥ ) xM> |() |fx A ε? < , 5?f ? ?f x→+∞ H[&1Kb:T lim ( ) x f xA →+∞ =  . () ( )fx Ax→→+∞ ó 0 x x→ H f ?￥K  !f ? ?()fx 0 x ￥ ? b? #× ( ) 0 0 ;Uxδ′ =μ?l &1? ? ? ?ó￥ 0, ( ) 0ε δδ′?> ?< > P¤? 0 0| |xx δ< ?< Hμ |() |fx A ε?< 5?f ? ? t?f x 0 x H[&1K ?&1 0 x x→ H ￥K :T()fx 0 lim ( ) xx f x → = A 0 () ( )f xAxx→→. ? ?§  K￥?l  !f ? f 0 0 (;)Uxδ + ′ =μ?l& 1? ?b ?  ?ó￥ 0, ( ) 0ε δδ′?> ?< > P¤? 00 xxxδ<< + Hμ |() |fx A ε? < , 5? ?&1 f ? ? t? f x 0 x H￥·K:T 0 lim ( ) xx f xA + → =  0 () ( )fx Ax x + →→ 0 (0)f xA+ = b ? ? Vó PK ?l 0 0 (;)Uxδ ?  00 x xxδ? <<  0 lim ( ) xx f x ? → = A 0 () ( )fx Ax x ? →→ 0 (0)f xA?= . = a '? ?  1. f ?K ￥?é? ? ò ·B?? ? ?K 0 lim ( ) xx f x → i5NK ^·B￥b ó ?μ? ?? ? ? 0 lim ( ) xx f x → i5 f 0 x ￥  b? #× =μ?b ? ?  |?? ? ? 0 lim ( ) 0 xx fx A → = > 5  ? ?? ? 0 rA< < i  P 0 0 ()Ux - 1 - "Q =SDyú ? [? ?Ds [?T Z? ¤B M 0 0 ()x Ux∈ μ  ?() 0fx r>> 0 lim ( ) 0 xx fx A → = < 5 ??μ ?  i  P¤B M 0Ar<< 0 0 ()Ux 0 0 ()x Ux∈ μ () 0fx r< <  ?  ?? T?? ? ! 0 lim ( ) xx f x → ? ?i  O  # × 0 lim ( ) xx gx → 0 0 (;)Uxδ′ =μ 5() ()fx gx≤ 00 lim ( ) lim ( ) xx xx f xg →→ ≤ xb ? ? ? ?? ?  ! 00 lim ( ) lim ( ) xx xx f xgx →→ A= =  O  0 0 (;)Uxδ′ =μ 5() () ()fx hx gx≤≤ 0 lim ( ) xx hx A → = b 2. f ?K ￥ ??? ? ò B2e5  ! f 0 0 (;)Uxδ′ = μ ? l  0 lim ( ) xx f x → i ? ??c? 0 0 (;)Uxδ′ O [ 0 x 1K￥ ?  { } n x K lim ( ) n n f x →∞ ?i OM?b ó !f ? f 0 x ￥  b? # × = μ ?l 0 0 ()Ux + 0 lim ( ) xx f xA + → =  ?? [? 0 x 1 K￥?h ?  { } 0 0 () n x Ux + ? μ lim ( ) n n f xA →∞ = . ? ?? μ?? ? ! 1?lμ ￥??μ?f ? 5·Kf 0 0 ()Ux + 0 lim ( ) xx f x + → ib ? O l ? 5 ! f ? f 0 0 (;)Uxδ′ =μ? l 0 lim ( ) xx f x → i ? ?ó 0ε >  i? ? ()δ δ′<  P¤ ?? 0 0 ,(;)xx U xδ′′′∈ ) ()|fx fxμ |( ε′ ′′? < b 3. ?×1 K ò 0 sin lim 1 x x x → = ó 1 lim 1 x x e x →∞ ?? += ?? ??  () 1 0 lim 1 e α α α → +=. ? a '1 p  1. ? ? ?3 ? K￥ N?ε ?liD?¨ ?￡ó?￥ ? Kb 2. g? ?  K￥?éi ?¨ ?￡ ü9 ?ó?￥ ? Kb 3. g? ?  Ki ￥  1H q?  sH q i ? ¨? tHq ￡ ü  ? ?   K ￥i?b - 2 - "Q =SDyú ? [? ?Ds [?T Z? 4. ? g? ×1K e n n n = ? ? ? ? ? ? + ∞→ 1 1lim i ?¨ ?9 ? t ? ￥Kb 1 a? ? è5  è 1. ! ? ? ? ? ? ? ? = 1 1 sin )( x x xf  ksY p 0 0 < > x x )00( +f ? i) ? ￥i?b )00( ?f s f ? ^?s f ?? ?0>x 0<x H Vr T? M ] p P K?·  K H 1 P¨M?￥Vr T' x xf x 1 sinlim)00( 0 + → =+  1lim)00( 0 ? → =? x f b 3 ?? Hμ0>x x x x << 1 sin0 #?K ￥ ? ? ?? 0 1 sinlim)00( 0 ==+ + → x xf x  6BZ ? 11lim)00( 0 ==? ? →x f ? ^  #  ￥K?ib )00()00( ?≠+ ff )(xf 0=x è 2. p/  Kò x x x ))nsin(sin(si lim 0→ ó x x nx nx ? ? ? ? ? ? ? + ∞→ lim  1? ? n 3 ò 1111 sin sin )sin(sin )sin(sin ))nsin(sin(si lim ))nsin(sin(si lim 00 =??=??= →→ x x x x x x x x xx ó ? ? ? ? ? ? ? ?? ? ∞→∞→∞→ ∞→ ? ? ? ? ? ? ? ? ? ? ? ? ? += ? ? ? ? ? ? ? += ? ? ? ? ? ? ? + nx x n n nx x x x x x x n nx nx n nx nx lim2 2 2 1 1lim 2 1limlim nn ee 212 == ? è 3 ￡ 5372 2 3 +? → 12933 lim 23 = ?+? xx xxx x   ￡ ü ?  3≠x ( )( ) ()() 5 12 12 3 5 12 312 33 5 12 372 933 22 2 23 ? ? + =? ?? ?+ =? +? ?+? x x xx xx xx xxx  - 3 - "Q =SDyú ? [? ?Ds [?T Z?  12 395 125 395 ? ?? ≤ ? ?? = x xx x xx   1 P 11635615595 ≤+?≤+?=? xxx  3μ 13 <?x    1 P 132556212 >??≥+?=? xxx 3μ 23 <?x   ? ^ K ? 130 <?< x  üμ   LL.311 12 395 5 12 372 933 2 23 ?≤ ? ?? ≤? +? ?+? x x xx xx xxx   è 4 0 1 2 lim 2 = ? ? ? ? ? ? ? ? ?? + ? ∞→ bax x x x  p a? b 3EB  ( ) ()∞→→ + +?+? = + ??? =? + ? xb x axxa x axaxx ax x x , 1 21 1 2 1 2 2222  ;1,01 ?==+? aa ? 1, =?=? bba    3E=  ? ? ? ? ? ? ? ? ?? + ? =?? + ? x b a xx x xbax x x 2 22 2 1 2 ? ∞→x Oe TKi ? 0 2 2 2 →?? + ? x b a xx x ' 1 2 lim 2 2 ?= ? ? ? ? ? ? ? ? ? + ? = ∞→ x b xx x a x  1 1 2 lim 2 = ? ? ? ? ? ? ? ? + + ? = ∞→ x x x b x  è 5 p ( )xxI x ?+= +∞→ 1lim 2 1 ? ( )xxI x ?+= ?∞→ 1lim 2 2 i a üK  ( )xx x ?+ ∞→ 1lim 2 ^?i 3 0 1 1 lim 2 1 = ++ = +∞→ xx I x  - 4 - "Q =SDyú ? [? ?Ds [?T Z? ( ) ∞=++= +∞→ ?= ttI t tx 1lim 2 2  VnK ( )xx x ?+ ∞→ 1lim 2 ?i ?aˉ 5  1¨K? l￡ ü/ K (1) 2 1 31 lim 29 x x x →? ? = ?  (2) 2 3 31 lim 69 x x x → ? = ?  (3) 1 (2)(1) lim 0 3 x xx x → ?? = ?  (4) 2 2 lim 5 3 x x → + =  (5) 2 1 (1)1 lim 21 x xx x → ? = ?  (6) 2 2 5 lim 1 1 x x x →∞ ? = ? . 2¨K￥ 15 ?E5 p/ K (1) 2 2 0 1 lim 21 x x xx → ? ? ?  (2) 2 2 1 1 lim 21 x x xx → ? ? ?  (3) 3 23 0 (1)(13) lim 2 x x x xx → ? + ? +  (4) 2 2 3 56 lim x xx xx → ? + ? 8 + 15  (5) 1 1 lim 1 n m x x x → ? ?  1?? ?  ,nm (6) 4 12 3 lim 2 x x x → + ? ? . 3 ! ￡ ü ?() 0fx > 0 lim ( ) xx f xA → = 5 0 lim ( ) n n xx f xA → =  ??? ? . n ≥ 2 4￡ ü ? 0 lim ( ) xx f xA → = 5 0 lim | ( ) | | | xx f xA → =?Q-?? . 5 p/ f ?3 ? U?￥P·K (1)  2 1, () 1, 2, 1, x fx x xx ? 0 , > ? = 1 , = ? ? + < ? =1x  - 5 - "Q =SDyú ? [? ?Ds [?T Z? (2) 2 1 sin , () , xx fx x xx ? , > 0 ? = ? ? 1+ , < 0 ?  =0x  (3) 2 || 1 () , 1 x fx x x = +  =0x  (4)  2 , () 0, ,0 x x fx x xx ? 2 , > 0 ? = 0 , = ? ? 1+ < ? , =x 0 . 6 p/  K (1) 2 2 1 lim 21 x x xx →∞ ? ? ?  (2) 57 lim 2 x x x x →+∞ ? +  (3) 2 lim ( 1 x x x →+∞ + ? )  (4) 2 lim ( 1 x x x →?∞ + ? )  7¨M 9 D p/ K (1) 0 1 lim [ ] x x x + →  (2)  0 lim ln ( 0) a x xxa + → > (3) ln lim 0 a x x a x →+∞ ( > )  (4) 1 lim x x x →+∞ . 8 !   ? ?  6 ()fx (, )a +∞ lim n n x →∞ = +∞ ? lim ( ) n n fx A →∞ = p￡ lim ( ) x fx A →+∞ =  A V[1í k . 9 ! "?()fx X ?l5 ()fx  X í?￥ 1Hq ^i , n x X ∈ 1, 2,n = L P . lim ( ) | n fx →∞ | = +∞ 10 ?¨× 1K pK (1) 2 2 0 sin lim (sin ) x x x →  (2) 2 0 cos 5 cos 3 lim x x x x → ?  (3) 3 0 tan sin lim x x x x → ?  (4) 0 arctan lim x x x →  - 6 - "Q =SDyú ? [? ?Ds [?T Z? (5) 0 cos( arccos ) lim x nx n x → ( )1  ?  (6) 4 tan 1 lim 4 x x x π π → ? ?  (7) lim x x x ? →∞ 2?? 1 ?? ??   (8) 1 0 lim (1 ) x x nx n → + ( )1? ?  (9) 1 0 1 lim ( ) 1 x x x x → + ?  (10) 2 2 2 1 lim 1 x x x x →∞ ?? ? ?? ? ??  11￡ ü 0 1 lim cos x x → ?i . 12￡ ü 0 lim ( ) xx D x → ?i ? 1, () ,. x Dx x ? = ? 0 ? 1 μ ? ?  1í ? ? 13 pK lim cos cos cos 24 2 n n x xx →+∞ L . 14¨?l ￡ ü (1) ? lim ( ) xa fx → = +∞ lim ( ) xa g xA → = 5 lim ( ) ( )] xa fx gx → [ + = +∞  (2) ?  5lim ( ) xa fx → = +∞ lim ( ) xa gx A → = ( >0) lim ( ) ( )] xa fxgx → [ = +∞ . 15 ? lim ( ) x fx A →+∞ = lim ( ) x g xB →+∞ = ￡ ü lim ( ) ( )] x fxgx AB →+∞ [ = . 16￡ ü ￥ 1Hq ^ ?? ? lim ( ) x fx A →+∞ = )( n xn → +∞ →∞ μ (( n fx An ) → →∞) ) . 17￡ ü ￥ 1Hq ^ ?? ?  0 lim ( ) xx fx + → = +∞ 0 ( n xxn → →∞ μ (( n fx An ) → →∞) . 18 !f ?   ?@Z?()fx (0, ) +∞ (2 ) ( )fx fx =  O lim ( ) x fx A →+∞ = ￡ ü () , (0, )fx A x ≡ ∈ +∞ - 7 -