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❚r÷í♥❣ ✣❍ ❚➔✐ ❝❤➼♥❤✲▼❛r❦❡t✐♥❣
❑❤♦❛ ❈ì ❇↔♥

❈ë♥❣ ❤á❛ ❳➣ ❤ë✐ ❈❤õ ♥❣❤➽❛ ❱✐➺t ◆❛♠
✣ë❝ ❧➟♣✲❚ü ❞♦✲❍↕♥❤ ♣❤ó❝

✣➋ ❚❍■ ❙➮ ✶ ▼➷◆ ❚❖⑩◆ ❈❆❖ ❈❻P ✭❈▲❈✮
❚❤í✐ ❣✐❛♥✿ ✼✺ ♣❤ót ✭❦❤æ♥❣ ❦➸ ❝❤➨♣✴♣❤→t ✤➲✮
✶✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ ❣✐î✐ ❤↕♥

1

lim (cos(x)) x2

x→0

✷✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ t➼❝❤ ♣❤➙♥ s✉② rë♥❣
+∞
2

1


dx
x x2 − 1

✸✳ ✭✶✱✺ ✤✐➸♠✮ ❙û ❞ö♥❣ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ✤➸ t➼♥❤ ❣➛♥ ✤ó♥❣ ❣✐→ trà ❜✐➸✉ t❤ù❝ s❛✉
(2, 99)2 + (4, 02)2

A=

✹✳ ✭✷✱✵ ✤✐➸♠✮ ❚➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ ❤❛✐ ❜✐➳♥ s❛✉
f (x, y) = 2x2 + y 2 − 4x + 8

✺✳ ✭✷✱✵ ✤✐➸♠✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✶ s❛✉
y − 2xy = x3

✻✳ ✭✶✱✺ ✤✐➸♠✮ ❈❤♦ ❤➔♠ sè
f (x) =




1
1

1+e x−2

, ✈î✐ x = 2
, ✈î✐ x = 2.

0

❍➔♠ sè ❢✭①✮ ❝â ❧✐➯♥ tö❝ t↕✐ ①❂✷ ❤❛② ❦❤æ♥❣❄

❈❤ó þ✿
✶✳ ❙✐♥❤ ✈✐➯♥ ❦❤æ♥❣ ✤÷ñ❝ t❤❛♠ ❦❤↔♦ t➔✐ ❧✐➺✉✳
✷✳ ❈→♥ ❜ë ❝♦✐ t❤✐ ❦❤æ♥❣ ❣✐↔✐ t❤➼❝❤ ❣➻ t❤➯♠✳
❇ë ♠æ♥ ❚♦→♥✲❚❤è♥❣ ❦➯ ✭❞✉②➺t✮

◆❣÷í✐ r❛ ✤➲





❚r÷í♥❣ ✣❍ ❚➔✐ ❝❤➼♥❤✲▼❛r❦❡t✐♥❣
❑❤♦❛ ❈ì ❇↔♥

❈ë♥❣ ❤á❛ ❳➣ ❤ë✐ ❈❤õ ♥❣❤➽❛ ❱✐➺t ◆❛♠
✣ë❝ ❧➟♣✲❚ü ❞♦✲❍↕♥❤ ♣❤ó❝

✣➋ ❚❍■ ❙➮ ✷ ▼➷◆ ❚❖⑩◆ ❈❆❖ ❈❻P ✭❈▲❈✮
❚❤í✐ ❣✐❛♥✿ ✼✺ ♣❤ót ✭❦❤æ♥❣ ❦➸ ❝❤➨♣✴♣❤→t ✤➲✮
✶✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ ❣✐î✐ ❤↕♥
x+2
x−3

lim

x→±∞

3x+4

✷✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ t➼❝❤ ♣❤➙♥ s✉② rë♥❣
+∞

e−x sin(x)dx
0

✸✳ ✭✶✱✺ ✤✐➸♠✮ ❙û ❞ö♥❣ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ✤➸ t➼♥❤ ❣➛♥ ✤ó♥❣ ❣✐→ trà ❜✐➸✉ t❤ù❝ s❛✉
A=

(1, 99)2 + (3, 02)2

✹✳ ✭✷✱✵ ✤✐➸♠✮ ❚➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ ❤❛✐ ❜✐➳♥ s❛✉
f (x, y) = 4x + 2y − x2 − y 2

✺✳ ✭✷✱✵ ✤✐➸♠✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✶ s❛✉
y + 4y = x2 e−4x

✻✳ ✭✶✱✺ ✤✐➸♠✮ ❈❤♦ ❤➔♠ sè
f (x) =

sin3 (x)
x

, ✈î✐ x = 0

a

, ✈î✐ x = 0.

❳→❝ ✤à♥❤ ❛ ✤➸ ❤➔♠ sè ❢✭①✮ ❧✐➯♥ tö❝ t↕✐ ①❂✵✳

❈❤ó þ✿
✶✳ ❙✐♥❤ ✈✐➯♥ ❦❤æ♥❣ ✤÷ñ❝ t❤❛♠ ❦❤↔♦ t➔✐ ❧✐➺✉✳
✷✳ ❈→♥ ❜ë ❝♦✐ t❤✐ ❦❤æ♥❣ ❣✐↔✐ t❤➼❝❤ ❣➻ t❤➯♠✳
❇ë ♠æ♥ ❚♦→♥✲❚❤è♥❣ ❦➯ ✭❞✉②➺t✮

◆❣÷í✐ r❛ ✤➲




❚r÷í♥❣ ✣❍ ❚➔✐ ❝❤➼♥❤✲▼❛r❦❡t✐♥❣
❑❤♦❛ ❈ì ❇↔♥

❈ë♥❣ ❤á❛ ❳➣ ❤ë✐ ❈❤õ ♥❣❤➽❛ ❱✐➺t ◆❛♠
✣ë❝ ❧➟♣✲❚ü ❞♦✲❍↕♥❤ ♣❤ó❝

✣➋ ❚❍■ ❙➮ ✸ ▼➷◆ ❚❖⑩◆ ❈❆❖ ❈❻P ✭❈▲❈✮
❚❤í✐ ❣✐❛♥✿ ✼✺ ♣❤ót ✭❦❤æ♥❣ ❦➸ ❝❤➨♣✴♣❤→t ✤➲✮
✶✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ ❣✐î✐ ❤↕♥

sin(x)
x−sin(x)

sin(x)
x

lim

x→0

✷✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ t➼❝❤ ♣❤➙♥ s✉② rë♥❣
+∞

e−x cos(x)dx
0

✸✳ ✭✶✱✺ ✤✐➸♠✮ ❙û ❞ö♥❣ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ✤➸ t➼♥❤ ❣➛♥ ✤ó♥❣ ❣✐→ trà ❜✐➸✉ t❤ù❝ s❛✉
(3, 99)2 + (5, 02)2

A=

✹✳ ✭✷✱✵ ✤✐➸♠✮ ❚➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ ❤❛✐ ❜✐➳♥ s❛✉
f (x, y) = (x + y − 9)(4x + 3y) − 6xy

✺✳ ✭✷✱✵ ✤✐➸♠✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✶ s❛✉
xy + 2y = x−3

✻✳ ✭✶✱✺ ✤✐➸♠✮ ❈❤♦ ❤➔♠ sè
f (x) =




1
1

x+2 x−3

, ✈î✐ x = 3
, ✈î✐ x = 3.

a

❳→❝ ✤à♥❤ ❛ ✤➸ ❤➔♠ sè ❢✭①✮ ❧✐➯♥ tö❝ t↕✐ ①❂✸✳

❈❤ó þ✿
✶✳ ❙✐♥❤ ✈✐➯♥ ❦❤æ♥❣ ✤÷ñ❝ t❤❛♠ ❦❤↔♦ t➔✐ ❧✐➺✉✳
✷✳ ❈→♥ ❜ë ❝♦✐ t❤✐ ❦❤æ♥❣ ❣✐↔✐ t❤➼❝❤ ❣➻ t❤➯♠✳
❇ë ♠æ♥ ❚♦→♥✲❚❤è♥❣ ❦➯ ✭❞✉②➺t✮

◆❣÷í✐ r❛ ✤➲




❚r÷í♥❣ ✣❍ ❚➔✐ ❝❤➼♥❤✲▼❛r❦❡t✐♥❣
❑❤♦❛ ❈ì ❇↔♥

❈ë♥❣ ❤á❛ ❳➣ ❤ë✐ ❈❤õ ♥❣❤➽❛ ❱✐➺t ◆❛♠
✣ë❝ ❧➟♣✲❚ü ❞♦✲❍↕♥❤ ♣❤ó❝

✣➋ ❚❍■ ❙➮ ✹ ▼➷◆ ❚❖⑩◆ ❈❆❖ ❈❻P ✭❈▲❈✮
❚❤í✐ ❣✐❛♥✿ ✼✺ ♣❤ót ✭❦❤æ♥❣ ❦➸ ❝❤➨♣✴♣❤→t ✤➲✮
✶✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ ❣✐î✐ ❤↕♥
lim

x→0

ln cos(x)
x2

✷✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ t➼❝❤ ♣❤➙♥ s✉② rë♥❣
+∞

2

x3 e−x dx
0

✸✳ ✭✶✱✺ ✤✐➸♠✮ ❙û ❞ö♥❣ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ✤➸ t➼♥❤ ❣➛♥ ✤ó♥❣ ❣✐→ trà ❜✐➸✉ t❤ù❝ s❛✉
A = ln (2, 01)2 + (3, 99)2

✹✳ ✭✷✱✵ ✤✐➸♠✮ ❚➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ ❤❛✐ ❜✐➳♥ f (x, y) = 2x2 −6y 2 , ✈î✐ r➔♥❣ ❜✉ë❝ x+2y = 6.
✺✳ ✭✷✱✵ ✤✐➸♠✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉
y ” − 6y + 9y = 2x2 − x + 3

✻✳ ✭✶✱✺ ✤✐➸♠✮ ❈❤♦ ❤➔♠ sè
f (x) =




1
1

x+2 x−1

, ✈î✐ x = 1
, ✈î✐ x = 1.

0

❍➔♠ sè ❢✭①✮ ❝â ❧✐➯♥ tö❝ ✈î✐ ♠å✐ x ∈ (−∞, +∞) ❤❛② ❦❤æ♥❣❄✳

❈❤ó þ✿
✶✳ ❙✐♥❤ ✈✐➯♥ ❦❤æ♥❣ ✤÷ñ❝ t❤❛♠ ❦❤↔♦ t➔✐ ❧✐➺✉✳
✷✳ ❈→♥ ❜ë ❝♦✐ t❤✐ ❦❤æ♥❣ ❣✐↔✐ t❤➼❝❤ ❣➻ t❤➯♠✳
❇ë ♠æ♥ ❚♦→♥✲❚❤è♥❣ ❦➯ ✭❞✉②➺t✮

◆❣÷í✐ r❛ ✤➲




❚r÷í♥❣ ✣❍ ❚➔✐ ❝❤➼♥❤✲▼❛r❦❡t✐♥❣
❑❤♦❛ ❈ì ❇↔♥

❈ë♥❣ ❤á❛ ❳➣ ❤ë✐ ❈❤õ ♥❣❤➽❛ ❱✐➺t ◆❛♠
✣ë❝ ❧➟♣✲❚ü ❞♦✲❍↕♥❤ ♣❤ó❝

✣➋ ❚❍■ ❙➮ ✺ ▼➷◆ ❚❖⑩◆ ❈❆❖ ❈❻P ✭❈▲❈✮
❚❤í✐ ❣✐❛♥✿ ✼✺ ♣❤ót ✭❦❤æ♥❣ ❦➸ ❝❤➨♣✴♣❤→t ✤➲✮
✶✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ ❣✐î✐ ❤↕♥
1−

lim

x→0

cos(x)
x2

✷✳ ✭✶✱✺ ✤✐➸♠✮ ❚➼♥❤ t➼❝❤ ♣❤➙♥ s✉② rë♥❣
+∞

e−x

0

1
dx
+ ex

✸✳ ✭✶✱✺ ✤✐➸♠✮ ❙û ❞ö♥❣ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ✤➸ t➼♥❤ ❣➛♥ ✤ó♥❣ ❣✐→ trà ❜✐➸✉ t❤ù❝ s❛✉
A = ln (3, 01)2 + (4, 99)2

✹✳ ✭✷✱✵ ✤✐➸♠✮ ❚➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ ❤❛✐ ❜✐➳♥ f (x, y) = x2 + 3xy − 5y 2 , ✈î✐ r➔♥❣ ❜✉ë❝
2x + 3y = 6.

✺✳ ✭✷✱✵ ✤✐➸♠✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉
y ” − 9y + 20y = x2 e4x

✻✳ ✭✶✱✺ ✤✐➸♠✮ ❈❤♦ ❤➔♠ sè
f (x) =




1
1

x+3 x−2

, ✈î✐ x = 2
, ✈î✐ x = 2.

0

❍➔♠ sè ❢✭①✮ ❝â ❧✐➯♥ tö❝ ✈î✐ ♠å✐ x ∈ (−∞, +∞) ❤❛② ❦❤æ♥❣❄

❈❤ó þ✿
✶✳ ❙✐♥❤ ✈✐➯♥ ❦❤æ♥❣ ✤÷ñ❝ t❤❛♠ ❦❤↔♦ t➔✐ ❧✐➺✉✳
✷✳ ❈→♥ ❜ë ❝♦✐ t❤✐ ❦❤æ♥❣ ❣✐↔✐ t❤➼❝❤ ❣➻ t❤➯♠✳
❇ë ♠æ♥ ❚♦→♥✲❚❤è♥❣ ❦➯ ✭❞✉②➺t✮

◆❣÷í✐ r❛ ✤➲





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