7.2. Gi´o . iha . n h`am mˆo . tbiˆe ´ n 31 v`a c´ac hˆe . qua ’ cu ’ a (7.13) lim x→∞  1+ 1 x  x = e, (7.14) lim x→0 log a (1 + x) x = 1 lna , 0 <a=1, (7.15) lim x→0 a x − 1 x =lna, 0 <a=1. (7.16) C ´ AC V ´ IDU . V´ı du . 1. Su . ’ du . ng (ε −δ)-d i . nh ngh˜ıa gi´o . iha . nd ˆe ’ ch´u . ng minh r˘a ` ng lim x→−3 x 2 =9. Gia ’ i. Ta cˆa ` nch´u . ng minh r˘a ` ng ∀ε>0, ∃δ>0 sao cho v´o . i |x +3| <δth`ı ta c´o |x 2 − 9| <ε. Ta cˆa ` nu . ´o . clu . o . . ng hiˆe . u |x 2 − 9|. ta c´o |x 2 − 9| = |x −3||x +3|. Do th`u . asˆo ´ |x −3| khˆong bi . ch˘a . n trˆen to`an tru . csˆo ´ nˆen dˆe ’ u . ´o . clu . o . . ng t´ıch do . n gia ’ nho . n ta tr´ıch ra 1 - lˆan cˆa . ncu ’ adiˆe ’ m a = −3t´u . cl`a khoa ’ ng (−4; −2). V´o . imo . i x ∈ (−4; −2) ta c´o |x − 3| < 7v`adod´o |x 2 − 9| < 7|x +3|. V`ı δ-lˆan cˆa . ndiˆe ’ m a = −3[t´u . c l`a khoa ’ ng (−3 − δ; −3+δ)] khˆong d u . o . . cvu . o . . t ra kho ’ i ranh gi´o . icu ’ a 1-lˆan cˆa . n nˆen ta lˆa ´ y δ = min  1, ε 7  . Khi d ´ov´o . i0< |x +3| <δ⇒|x 2 − 9| <ε. Do vˆa . y lim x→−3 x 2 =9.  V´ı du . 2. Ch´u . ng minh r˘a ` ng lim x→2 √ 11 − x =3. Gia ’ i. Gia ’ su . ’ ε>0 l`a sˆo ´ du . o . ng cho tru . ´o . cb´e bao nhiˆeu t`uy ´y. Ta x´et bˆa ´ tphu . o . ng tr`ınh | √ 11 − x − 3| <ε. (7.17) 32 Chu . o . ng 7. Gi´o . iha . n v`a liˆen tu . ccu ’ a h`am sˆo ´ Ta c´o (7.17) ⇔−ε< √ 11 − x − 3 <ε⇔    √ 11 − x − 3 > −ε √ 11 − x − 3 <ε ⇔    x − 11 < −(3 − ε) 2 x − 11 > −(3 + ε) 2 ⇔    x − 2 < 6ε − ε 3 x − 2 > −(6ε + ε 2 ). V`ı6ε − ε 2 < |−(6ε + ε) 2 | =6ε + ε 2 nˆen ta c´o thˆe ’ lˆa ´ y δ(ε) l`a sˆo ´ δ  6ε − ε 2 .V´o . isˆo ´ δ d´o ta thˆa ´ yr˘a ` ng khi x tho ’ a m˜an bˆa ´ td˘a ’ ng th´u . c 0 < |x − 2| <δth`ı | √ 11 − x − 3| <εv`a lim x→2 √ 11 − x =3.  V´ı d u . 3. T´ınh c´ac gi´o . iha . n 1) lim x→2 2 x − x 2 x − 2 (vˆo di . nh da . ng 0 0 ); 2) lim x→ π 4 cotg2x · cotg  π 4 − x  (vˆo di . nh da . ng 0 ·∞); 3) lim x→∞  e 1 x + 1 x  x (vˆo di . nh da . ng 1 ∞ ). Gia ’ i 1) Ta c´o 2 x − x 2 x −2 = 2 x −2 2 −(x 2 − 2 2 ) x − 2 =4· 2 x−2 − 1 x −2 − x 2 − 4 x − 2 · T`u . d ´o suy r˘a ` ng lim x→2 2 x − x 2 x − 2 = 4 lim x→2 2 x−2 − 1 x − 2 − lim x→2 x 2 −4 x − 2 = 4ln2 − 4. 2) D˘a . t y = π 4 −x. Khi d´o lim x→ π 4 cotg2x · cotg  π 4 − x  = lim y→0 cotg  π 2 − 2y  cotgy = lim y→0 sin 2y sin y · cos y cos 2y =2. 7.2. Gi´o . iha . n h`am mˆo . tbiˆe ´ n 33 3) D˘a . t y = 1 x . Khi d´o lim x→∞  e 1 x + 1 x  x = lim y→0 (e y + y) 1 y = e lim y→0 ln (e y + y) y ; lim y→0 ln(e y + y) y = lim y→0 ln[1 + (e y + y −1)] e y + y − 1 · e y + y − 1 y = lim t→0 ln(1 + t) t · lim y→0  1+ e y − 1 y  =2. T`u . d´o suy r˘a ` ng lim y→0  e y + y  1 y = e 2 .  V´ı du . 4. Ch´u . ng to ’ r˘a ` ng h`am f( x) = sin 1 x khˆong c´o gi´o . iha . n khi x → 0. Gia ’ i. Ta lu . u´ymˆe . nh d ˆe ` phu ’ di . nh dˆo ´ iv´o . id i . nh ngh˜ıa gi´o . iha . n: lim x→a f(x) = A ⇔∃ε 0 > 0 ∀δ>0 ∃x δ (0 < |x δ − a| <δ) →|f(x 0 ) − A|  ε 0 . Nˆe ´ u A =0talˆa ´ y ε 0 = 1 2 v`a x k = 2 π 2 +2kπ . Khi d´o ∀δ>0, ∃k ∈ N :0<x k <δv`a |f(x k ) − 0| = |f(x k )| =1>ε 0 v`a nhu . vˆa . y A = 0 khˆong pha ’ i l`a gi´o . iha . ncu ’ ah`amd ˜a cho khi x → 0. Nˆe ´ u A = 0 th`ı ta lˆa ´ y ε 0 = |A| 2 v`a x k = 1 2kπ . Khi d´o ∀δ>0, ∃k ∈ N :0<x k <δth`ı |f(x k ) − A| = |A| >ε.Nhu . vˆa . ymo . isˆo ´ A =0d ˆe ` u khˆong l`a gi´o . iha . ncu ’ a h`am sin 1 x khi x → 0.  V´ı du . 5. H`am Dirichlet D( x): D(x)=    1nˆe ´ u x ∈ Q, 0nˆe ´ u x ∈ R \Q 34 Chu . o . ng 7. Gi´o . iha . n v`a liˆen tu . ccu ’ a h`am sˆo ´ khˆong c´o gi´o . iha . nta . i ∀a ∈ R. Gia ’ i. Ta ch´u . ng minh r˘a ` ng ta . imo . idiˆe ’ m a ∈ R h`am D(x) khˆong tho ’ a m˜an D i . nh l´y 2. Dˆe ’ l`am viˆe . cd´o, ta chı ’ cˆa ` nchı ’ ra hai d˜ay (a n )v`a (a  n )c`ung hˆo . itu . dˆe ´ n a sao cho lim n→∞ D(a n ) = lim n→∞ D(a  n ). D ˆa ` u tiˆen ta x´et d˜ay c´ac diˆe ’ mh˜u . uty ’ (a n )hˆo . itu . dˆe ´ n a.Tac´o D(a n )=1∀n v`a do d´o lim n→∞ D(a n ) = 1. Bˆay gi`o . ta x´et d˜ay (a  n )- d˜ay c´ac d iˆe ’ mvˆoty ’ hˆo . itu . dˆe ´ n a.Tac´oD(a  n )=0∀n v`a do vˆa . y lim n→∞ D(a  n )=0. Nhu . vˆa . y lim n→∞ D(a n ) = lim n→∞ D(a  n ). T`u . d ´o suy ra r˘a ` ng ta . idiˆe ’ m a h`am D(x) khˆong c´o gi´o . iha . n. V´ı d u . 6. Gia ’ su . ’ lim x→a f(x)=b, lim x→a g(x)=+∞.Ch´u . ng minh r˘a ` ng lim x→a [f(x)+g(x)] = +∞. Gia ’ i. Ta cˆa ` nch´u . ng minh r˘a ` ng ∀M>0, ∃δ>0 sao cho ∀x :0< |x − a| <δth`ı f(x)+g(x) >M. V`ı lim x→a f(x)=b nˆen tˆo ` nta . i δ 1 -lˆan cˆa . n U(a, δ 1 )cu ’ adiˆe ’ m a sao cho |f(x)| <C, x= a (7.18) trong d ´o C l`a h˘a ` ng sˆo ´ du . o . ng n`ao d ´o . Gia ’ su . ’ M>0 l`a sˆo ´ cho tru . ´o . ct`uy ´y. V`ı lim x→a g(x)=+∞ nˆen d ˆo ´ i v´o . isˆo ´ M + C, ∃δ>0(δ  δ 1 ) sao cho ∀x :0< |x − a| <δth`ı g(x) >M+ C (7.19) T`u . c´ac bˆa ´ td˘a ’ ng th´u . c (7.18) v`a(7.19) ta thu du . o . . c l`a: v´o . i x tho ’ a m˜an diˆe ` ukiˆe . n0< |x −a| <δ δ 1 th`ı f(x)+g(x)  g(x) −|f(x)| >M+ C − C = M.  B ` AI T ˆ A . P 7.2. Gi´o . iha . n h`am mˆo . tbiˆe ´ n 35 1. Su . ’ du . ng d i . nh ngh˜ıa gi´o . iha . n h`am sˆo ´ d ˆe ’ ch´u . ng minh c´ac d ˘a ’ ng th ´u . c sau d ˆay: 1) lim x→ π 6 sin x = 1 2 ; 2) lim x→ π 2 sin x =1; 3) lim x→0 x sin 1 x = 0; 4) lim x→+∞ arctgx = π 2 . Chı ’ dˆa ˜ n. D`ung hˆe . th ´u . c π 2 − arctgx<tg  π 2 − arctgx  = 1 x ) 5) lim x→∞ x −1 3x +2 = 1 3 ; 6) lim x→+∞ log a x =+∞; 7) lim x→+∞  √ x 2 +1− x  = 0; 8) lim x→−5 x 2 +2x − 15 x +5 = −8; 9) lim x→1 (5x 2 − 7x + 6) = 4; 10) lim x→2 x 2 − 3x +2 x 2 + x − 6 = 1 5 ; 11) lim x→+∞ x sin x x 2 − 100x + 3000 =0. 2. Ch´u . ng minh c´ac gi´o . iha . n sau dˆay khˆong tˆo ` nta . i: 1) lim x→1 sin 1 x −1 ; 2) lim x→∞ sin x; 3) lim x→o 2 1 x ; 4) lim x→0 e 1 x ; 5) lim x→∞ cos x. Nˆe ´ utu . ’ sˆo ´ v`a mˆa ˜ usˆo ´ cu ’ a phˆan th ´u . ch˜u . uty ’ dˆe ` u triˆe . t tiˆeu ta . idiˆe ’ m x = a th`ı c´o thˆe ’ gia ’ nu . ´o . c phˆan th´u . cchox − a (= 0) mˆo . t ho˘a . cmˆo . t sˆo ´ lˆa ` n. Su . ’ du . ng phu . o . ng ph´ap gia ’ nu . ´o . cd´o, h˜ay t´ınh c´ac gi´o . iha . n sau dˆay (3-10). 3. lim x→7 2x 2 − 11x − 21 x 2 − 9x +14 (DS. 17 5 ) 4. lim x→1 x 4 −x 3 + x 2 − 3x +2 x 3 −x 2 − x +1 (DS. 2) 5. lim x→1 x 4 +2x 2 −3 x 2 −3x +2 (DS. −8) 6. lim x→1 x m − 1 x n − 1 ; m, n ∈ Z (D S. m n ) 36 Chu . o . ng 7. Gi´o . iha . n v`a liˆen tu . ccu ’ a h`am sˆo ´ 7. lim x→1  1 1 − x − 3 1 − x 3  (DS. −1) 8. lim x→1  a 1 − x a − b 1 − x b  ; a, b ∈ N (D S. a −b 2 ) 9. lim x→1 (x n − 1)(x n−1 − 1) ···(x n−k+1 −1) (x − 1)(x 2 − 1) ···(x k − 1) (DS. C k n ) 10. lim x→a (x n − a n ) − na n−1 (x − a) (x −a) 2 , n ∈ N (DS. n(n − 1) 2 a n−1 ) Chı ’ dˆa ˜ n. Dˆo ’ ibiˆe ´ n x −a = t. C´ac b`ai to´an sau dˆay c´o thˆe ’ du . avˆe ` da . ng trˆen nh`o . ph´ep dˆo ’ ibiˆe ´ n (11-14) 11. lim x→1 x p q − 1 x r s − 1 (D S. ps qr ) 12. lim x→−1 1+ 3 √ x 1+ 5 √ x (DS. 5 3 ) 13. lim x→0 3 3 √ 1+x −4 4 √ 1+x +1 2 − 2 √ 1+x + x (DS. 1 6 ) 14. lim x→0 n √ 1+x − 1 x (DS. 1 n ) Mˆo . t trong c´ac phu . o . ng ph´ap t´ınh gi´o . iha . ncu ’ a c´ac biˆe ’ uth´u . cvˆoty ’ l`a chuyˆe ’ nvˆoty ’ t`u . mˆa ˜ usˆo ´ lˆen tu . ’ sˆo ´ ho˘a . c ngu . o . . cla . i (15-26) 15. lim x→0 √ 1+x + x 2 − 1 x (DS. 1 2 ) 16. lim x→2 √ 3+x + x 2 − √ 9 − 2x + x 2 x 2 − 3x +2 (DS. 1 2 ) 17. lim x→0 5x 3 √ 1+x − 3 √ 1 − x (D S. 15 2 ) 18. lim x→0 3 √ 1+3x − 3 √ 1 − 2x x + x 2 (DS. 2) 19. lim x→∞ √ x 2 +1− √ x 2 − 1  (DS. 0) 7.2. Gi´o . iha . n h`am mˆo . tbiˆe ´ n 37 20. lim x→∞  3 √ 1 − x 3 + x  (DS. 0) 21. lim x→+∞  √ x 2 +5x + x  (DS. +∞) 22. lim x→−∞ √ x 2 +5x + x  (DS. − 5 2 ) 23. lim x→+∞ √ x 2 +2x − x  (DS. 1) 24. lim x→−∞ √ x 2 +2x − x  .(DS. +∞) 25. lim x→∞  (x +1) 2 3 − (x −1) 2 3  (D S. 0) 26. lim x→+∞  n  (x + a 1 )(x + a 2 ) ···(x + a n ) − x  (D S. a 1 + a 2 + ···+ a n n ) Khi gia ’ i c´ac b`ai to´an sau dˆay ta thu . `o . ng su . ’ du . ng hˆe . th ´u . c lim t→0 (1 + t) α − 1 t = α (27-34) 27. lim x→0 5 √ 1+3x 4 − √ 1 − 2x 3 √ 1+x − √ 1+x (DS. −6) 28. lim x→0 n √ a + x − n √ a − x x , n ∈ N (DS. 2 n a 1 n −1 ) 29. lim x→0 √ 1+3x + 3 √ 1+x − 5 √ 1+x − 7 √ 1+x 4 √ 1+2x + x − 6 √ 1+x (D S. 313 280 ) 30. lim x→0 3 √ a 2 + ax + x 2 − 3 √ a 2 − ax + x 2 √ a + x − √ a −x (D S. 3 2 a 1 6 ) 31. lim x→0  √ 1+x 2 + x  n −  √ 1+x 2 − x  n x (DS. 2n) 32. lim x→0 n √ a + x − n √ a − x x , n ∈ N, a>0(DS. 2 n √ a na ) 33. lim x→0 n √ 1+ax − k √ 1+bx x , n ∈ N, a>0(DS. ak −bn nk ) 34. lim x→∞  n  (1 + x 2 )(2 + x 2 ) ···(n + x 2 ) − x 2  (D S. n +1 2 ) 38 Chu . o . ng 7. Gi´o . iha . n v`a liˆen tu . ccu ’ a h`am sˆo ´ Khi t´ınh gi´o . iha . n c´ac biˆe ’ uth´u . clu . o . . ng gi´ac ta thu . `o . ng su . ’ du . ng cˆong th ´u . cco . ba ’ n lim x→0 sin x x =1 c`ung v´o . isu . . kˆe ´ tho . . p c´ac phu . o . ng ph´ap t`ım gi´o . iha . nd ˜a n ˆeu o . ’ trˆen (35-56). 35. lim x→∞ sin πx 2 x (DS. 0) 36. lim x→∞ arctgx 2x (DS. 0) 37. lim x→−2 x 2 − 4 arctg(x +2) (DS. −4) 38. lim x→0 tgx − sin x x 3 (DS. 1 2 ) 39. lim x→0 xcotg5x (DS. 1 5 ) 40. lim x→1 (1 − x)tg πx 2 (DS. 2 π ) 41. lim x→1 1 − x 2 sin πx (DS. 2 π ) 42. lim x→π sin x π 2 − x 2 (DS. 1 2π ) 43. lim x→0 cos mx − cos nx x 2 (DS. 1 2 (n 2 − m 2 )) 44. lim x→∞ x 2  cos 1 x − cos 3 x  (DS. 4) 45. lim x→0 sin(a + x) + sin(a −x) −2sin a x 2 (DS. −sin a) 46. lim x→0 cos(a + x) + cos(a −x) −2 cos a 1 − cos x (D S. −2 cos a) 47. lim x→∞  sin √ x 2 +1− sin √ x 2 − 1  (DS. 0) 7.2. Gi´o . iha . n h`am mˆo . tbiˆe ´ n 39 48. lim x→0 √ cos x − 1 x 2 (DS. − 1 4 ) 49. lim x→ π 2 cos x 2 − sin x 2 cos x (D S. 1 √ 2 ) 50. lim x→ π 3 sin  x − π 3  1 − 2 cos x (D S. 1 √ 3 ) 51. lim x→ π 4 √ 2 cos x − 1 1 − tg 2 x (D S. 1 4 ) 52. lim x→0 √ 1+tgx − √ 1 − tgx sin x (DS. 1) 53. lim x→0 m √ cos αx − m √ cos βx x 2 (DS. β 2 − α 2 2m ) 54. lim x→0 cos x − 3 √ cos x sin 2 x (DS. − 1 3 ) 55. lim x→0 1 − cos x √ cos 2x tgx 2 (DS. 3 2 ) 56. lim x→0 √ 1+x sin x − cos x sin 2 x 2 (DS. 4) D ˆe ’ t´ınh gi´o . iha . n lim x→a [f(x)] ϕ(x) , trong d´o f(x) → 1, ϕ(x) →∞khi x → a ta c´o thˆe ’ biˆe ´ nd ˆo ’ ibiˆe ’ uth´u . c [f(x)] ϕ(x) nhu . sau: lim x→a [f(x)] ϕ(x) = lim x→a  [1 + (f(x) −1)] 1 f(x)−1  ϕ(x)[f(x)−1] = e lim x→a ϕ(x)[f(x)−1] o . ’ dˆay lim x→a ϕ(x)[f(x) −1] du . o . . c t´ınh theo c´ac phu . o . ng ph´ap d˜a nˆeu trˆen d ˆa y . N ˆe ´ u lim x→a ϕ(x)[f(x) −1] = A th`ı lim x→a [f(x)] ϕ(x) = e A (57-68). 40 Chu . o . ng 7. Gi´o . iha . n v`a liˆen tu . ccu ’ a h`am sˆo ´ 57. lim x→∞  2x +3 2x +1  x+1 (DS. e) 58. lim x→∞  x 2 − 1 x 2  x 4 (DS. 0) 59. lim x→0 (1 + tgx) cotgx (DS. e) 60. lim x→0 (1 + 3tg 2 x) cotg 2 x (DS. e 3 ) 61. lim x→0  cos x cos 2x  1 x 2 (DS. e 3 2 ) 62. lim x→ π 2 (sin x) 1 cotgx (DS. −1) 63. lim x→ π 2 (tgx) tg2x (DS. e −1 ) 64. lim x→0  tg  π 4 + x  cotg2x (DS. e) 65. lim x→0  cos x  1 x 2 (DS. e − 1 2 ) 66. lim x→0  cos 3x  1 sin 2 x (DS. e − 9 2 ) 67. lim x→0  1+tgx 1 + sin x  1 sin x (DS. 1) 68. lim x→ π 4  sin 2x  tg 2 2x (DS. e − 1 2 ) Khi t´ınh gi´o . iha . n c´ac biˆe ’ uth´u . cc´och´u . a h`am lˆodarit v`a h`am m˜uta thu . `o . ng su . ’ du . ng c´ac cˆong th´u . c (7.15) v`a (7.16) v`a c´ac phu . o . ng ph´ap t´ınh gi´o . iha . nd˜anˆeuo . ’ trˆen (69-76). 69. lim x→e lnx −1 x − e (DS. e −1 ) 70. lim x→10 lgx −1 x −10 (DS. 1 10ln10 ) 71. lim x→0 e x 2 − 1 √ 1 + sin 2 x −1 (DS. 2) 72. lim x→0 e x 2 − cos x sin 2 x (DS. 3 2 ) [...]... 3 Cho tru.´.c sˆ ε > 0 Theo dinh ngh˜a 1∗ ta o √ ´.c lu.o.ng mˆdun cua n´ Ta ’ a o o lˆp hiˆu f (x) − f (5) = x + 4 − 3 v` u o a e c´ o √ |x − 5| |x − 5| < | x + 4 − 3| = √ 3 | x + 4 + 3| (*) ´ Nˆu ta chon δ = 3 th` v´.i nh˜.ng gi´ tri x m` |x − 5| < δ = 3 e ı o u a a √ d´ suy r˘ng h`m f (x) liˆn tuc tai diˆm ’ ` u a a e e ta s˜ c´ | x + 4 − 3| < ε T` o e o x0 = 5 √ ` ’ e e a a V´ du 3. .. − 3) liˆn tuc ∀ x ∈ R ı u a a e ’ ´ ’ e u ´ e e Giai Ta lˆy diˆm x0 ∈ R t`y y X´t hiˆu a sin(2x − 3) − sin(2x0 − 3) = 2 cos(x + x0 − 3) sin(x − x0) = α(x) V` | cos(x + x0 − 3) | ı 1 v` sin(x − x0)| < |x − x0 | nˆn khi x → x0 a e ` ’ h`m sin(x − x0 ) l` h`m vˆ c`ng b´ T` d´ suy r˘ng α(x) l` t´ch cua a a a o u e u o a a ı e a h`m bi ch˘n v´.i vˆ c`ng b´ v` a a o o u lim sin(2x − 3) = sin(2x0 − 3) ...7 .3 H`m liˆn tuc a e eαx − eβx x→0 sin αx − sin βx 73 lim esin 5x − esin x x→0 ln(1 + 2x) 74 lim 2 41 (DS 1) (DS 2) 2 1 a ax − bx , a > 0, b > 0 (DS − ln ) 75 lim x→0 ln cos 2x 2 b sin x sin x 1 √ a +b x 76 lim , a > 0, b > 0 (DS ab) x→0 2 7 .3 H`m liˆn tuc a e - ’ ’ Dinh ngh˜ 7 .3. 1 H`m f (x) x´c dinh trong lˆn cˆn cua diˆm x0 ıa a a a a... ) ´ o u o V´.i ngˆn ng˜ sˆ gia dinh ngh˜a 7 .3. 1 c´ dang o ı o - ’ a a a a ’ e Dinh ngh˜ 7 .3. 1∗∗ H`m f (x) x´c dinh trong lˆn cˆn cua diˆm x0 ıa ´ du.o.c goi l` liˆn tuc tai x0 nˆu e a e lim ∆f = 0 ∆x→0 ´ ’ a Chu.o.ng 7 Gi´.i han v` liˆn tuc cua h`m sˆ o a e o 42 ` o B˘ng “ngˆn ng˜ d˜y” ta c´ dinh ngh˜a tu.o.ng du.o.ng a o u a ı - ’ Dinh ngh˜ 7 .3. 1∗∗∗ H`m f (x) x´c dinh trong lˆn cˆn diˆm... Dinh ngh˜ 7 .3. 1 H`m f (x) x´c dinh trong lˆn cˆn cua diˆm x0 ıa a a a a e ’ ´ du.o.c goi l` liˆn tuc tai diˆm d´ nˆu o e a e e lim f (x) = f (x0 ) x→x0 Dinh ngh˜ 7 .3. 1 tu.o.ng du.o.ng v´.i ıa o - ’ ’ a a a a e Dinh ngh˜ 7 .3. 1∗ H`m f (x) x´c dinh trong lˆn cˆn cua diˆm x0 ıa ’ ´ du.o.c goi l` liˆn tuc tai diˆm x0 nˆu e a e e ∀ ε > 0 ∃ δ > 0 ∀ x ∈ Df : |x − x0| < δ ⇒ |f (x) − f (x0 )| . − 6 √ 1+x (D S. 31 3 280 ) 30 . lim x→0 3 √ a 2 + ax + x 2 − 3 √ a 2 − ax + x 2 √ a + x − √ a −x (D S. 3 2 a 1 6 ) 31 . lim x→0  √ 1+x 2 + x  n −  √ 1+x 2 − x  n x (DS. 2n) 32 . lim x→0 n √ a. x 2 − 1 x (DS. 1 2 ) 16. lim x→2 √ 3+ x + x 2 − √ 9 − 2x + x 2 x 2 − 3x +2 (DS. 1 2 ) 17. lim x→0 5x 3 √ 1+x − 3 √ 1 − x (D S. 15 2 ) 18. lim x→0 3 √ 1+3x − 3 √ 1 − 2x x + x 2 (DS. 2) 19. lim x→∞ √ x 2 +1− √ x 2 −. gi´o . iha . n sau dˆay (3- 10). 3. lim x→7 2x 2 − 11x − 21 x 2 − 9x +14 (DS. 17 5 ) 4. lim x→1 x 4 −x 3 + x 2 − 3x +2 x 3 −x 2 − x +1 (DS. 2) 5. lim x→1 x 4 +2x 2 3 x 2 −3x +2 (DS. −8) 6. lim x→1 x m −