Limites Trigonometricos-Algebraicos _GRUPO 1

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ESCUELA SUPERIOR POLITÉCNICA DE CHIMBORAZO 
 
 
 
 
 
 
 
 
 
 
 
 
PRIMER SEMESTRE 
PARALELO ¨B¨ 
ANÁLISIS MATEMÁTICO 
2020-20 
ARIAS VELASTEGUI, SANTIAGO MATEO 
ASQUI VACA, BORIS JOSUE 
GONZALEZ LARA, ANDERSON JOSHUA 
MARFETAN SALINAS, LUIS ALEXANDER 
SANCHEZ CHUNZHO, GINNA JULAY 
 
 
 
 
 
LIMITES 
TRIGONOMÉTRICOS 
 
 
 
 
 
 
 
𝟏. 𝐥𝐢𝐦
𝒙→𝝅
𝟏−𝒔𝒆𝒏
𝒙
𝟐
𝝅−𝒙
 
Variable cambiada 𝑡 = 𝜋 − 𝑥 ; 𝑥 = 𝜋 + 𝑡 
lim
𝑡→0
1−𝑠𝑒𝑛(
𝜋
2
+
𝑡
2
)
−𝑡
→ lim
𝑡→0
1−𝑠𝑒𝑛(
𝜋
2
) cos(
𝑡
2
)+sen(
𝑡
2
) cos(
𝜋
2
)
−𝑡
 
=
1−cos(
𝑡
2
)+0 
− 𝑡
 
=lim
𝑡→0
1−cos(
𝑡
2
)
−𝑡
= lim
𝑡→0
[1−cos(
𝑡
2
)] [1+cos(
𝑡
2
)]
−𝑡[1+cos(
𝑡
2
)]
 
= lim
𝑡→0
 12−cos2(
𝑡
2
)
−𝑡[1+cos(
𝑡
2
)]
 
= lim
𝑡→0
 𝑠𝑒𝑛2(
𝑡
2
)
−𝑡[2]
→ lim
𝑡→0
 𝑠𝑒𝑛 (
𝑡
2
)
𝑡
2
∗2
∗
 𝑠𝑒𝑛 (
𝑡
2
)
−2 
 
= lim
𝑡→0
1
2
 𝑠𝑒𝑛 (
𝑡
2
)
−2
→
1
2
(0) 
= 0 
 
𝟐. 𝐥𝐢𝐦
𝒙→𝟎
𝐜𝐨𝐬 𝒙−𝐜𝐨𝐬𝟑𝒙 
𝒙𝟐
 
= lim
𝑥→0
cos𝑥−cos(2𝑥+𝑥)
𝑥2
→ lim
𝑥→0
cos𝑥−(cos2𝑥 cos𝑥−𝑠𝑒𝑛 2𝑥 𝑠𝑒𝑛 𝑥 )
𝑥2
 
= lim
𝑥→0
cos𝑥−cos2𝑥 cos𝑥+𝑠𝑒𝑛 2𝑥 𝑠𝑒𝑛 𝑥 )
𝑥2
 
= lim
𝑥→0
cos𝑥−(cos2 𝑥−𝑠𝑒𝑛2 𝑥) cos𝑥+2(𝑠𝑒𝑛 𝑥 cos𝑥) 𝑠𝑒𝑛 𝑥 
𝑥2
 
= lim
𝑥→0
cos𝑥−cos3 𝑥+𝑠𝑒𝑛2 𝑥 cos𝑥+2 𝑠𝑒𝑛2 𝑥 cos𝑥
𝑥2
 
= lim
𝑥→0
cos𝑥−cos3 𝑥+3𝑠𝑒𝑛2𝑥 cos𝑥 
𝑥2
 
= lim
𝑥→0
cos𝑥(1−cos2 𝑥+3 𝑠𝑒𝑛2 𝑥)
𝑥2
 
= lim
𝑥→0
cos𝑥(𝑠𝑒𝑛 2𝑥+3𝑠𝑒𝑛2 𝑥)
𝑥2
 
 
 
 
= lim
𝑥→0
cos𝑥(4𝑠𝑒𝑛 2𝑥)
𝑥2
 
= 4 ∗ lim
𝑥→0
cos 𝑥 ∗ lim
𝑥→0
 𝑠𝑒𝑛2 𝑥
𝑥2
 
= 4 ∗ lim
𝑥→0
cos 𝑥 ∗ lim
𝑥→0
𝑠𝑒𝑛 𝑥
𝑥
∗
𝑠𝑒𝑛 𝑥
𝑥
 
= 4cos(0) ∗ 1 ∗ 1 = 4 
 
 
𝟑. 𝐥𝐢𝐦
𝒙→𝟎
𝒕𝒂𝒏𝒈 𝒙−𝒔𝒆𝒏 𝒙
𝒙𝟑
 
= lim
𝑥→0
𝑠𝑒𝑛 𝑥 
cos𝑥
−
𝑠𝑒𝑛 𝑥
1
𝑥3
 
= lim
𝑥→0
𝑠𝑒𝑛𝑥−𝑠𝑒𝑛 𝑥cos𝑥
cos𝑥
𝑥3
1
 
= lim
𝑥→0
𝑠𝑒𝑛 𝑥−𝑠𝑒𝑛 𝑥∗cos𝑥 
𝑥3 cos𝑥 
 
= lim
𝑥→0
𝑠𝑒𝑛 𝑥(1−cos𝑥) 
𝑥3 cos𝑥 
∗
1+cos𝑥 
1+cos𝑥
 
= lim
𝑥→0
𝑠𝑒𝑛 𝑥(1−𝑐𝑜𝑠 2 𝑥) 
𝑥3 cos𝑥(1+cos𝑥) 
 
= lim
𝑥→0
𝑠𝑒𝑛 𝑥(𝑠𝑒𝑛 2𝑥)
𝑥3 cos𝑥(1+cos𝑥) 
 
=lim
𝑥→0
(𝑠𝑒𝑛 3𝑥)
𝑥3 cos𝑥(1+cos𝑥) 
 
= lim
𝑥→0
𝑠𝑒𝑛 3𝑥
𝑥3
∗
1
cos𝑥(1+cos𝑥)
 
= 1 ∗
1
cos(0)(1+cos0)
 
= 1 ∗
1
1(1+1)
 =
1
2
 
 
 
 
 
 
 
 
 
 
𝟒. 𝐥𝐢𝐦
𝒙→𝟎
𝒙 − 𝒔𝒆𝒏 𝟐𝒙
𝒙 + 𝒔𝒆𝒏 𝟑𝒙
 
= lim
𝑥→0
𝑥
𝑥
−
sen3x
x
𝑥
𝑥
+
sen2x
x
 
= lim
𝑥→0
1−
𝑠𝑒𝑛2𝑥
𝑥
1+
𝑐𝑜𝑠3𝑥
𝑥
 
= lim
𝑥→0
1−2𝑠𝑒𝑛2(0)
1+3cos3(0)
 
= lim
𝑥→0
1−2
1+3
 
= −
1
4
 
 
𝟓. 𝐥𝐢𝐦
𝒙→𝟎
𝟏 − √𝐜𝐨𝐬 𝒙
𝒙𝟐
 
= lim
𝑥→0
1−√cos𝑥
𝑥2
∗
1+√cos𝑥
1+√cos𝑥
 
= lim
𝑥→0
1−cos𝑥
𝑥2(1+√cos𝑥)
= 
= lim
𝑥→0
1−cos𝑥
2𝑥2
∗
1+cos𝑥
1+cos𝑥
= 
= lim
𝑥→0
12−𝑐𝑜𝑠 2𝑥
2𝑥2(2)
= 
=lim
𝑥→0
𝑠𝑒𝑛 2𝑥
4𝑥2
= 
= lim
𝑥→0
1
4
 (
𝑠𝑒𝑛 𝑥
𝑥
)
2
= 
=
1
4
 
 
𝟔. 𝐥𝐢𝐦
𝒙→𝟎
𝒔𝒆𝒏(𝒙 + 𝒉) − 𝒔𝒆𝒏𝒙
𝒉
 
= lim
ℎ→0
(𝑠𝑒𝑛 𝑥)(cosℎ) +(cos𝑥)(𝑠𝑒𝑛 ℎ)
ℎ
 
 
 
 
 
= lim
ℎ→0
(𝑠𝑒𝑛 𝑥∗cosℎ−𝑠𝑒𝑛 𝑥) + cosx senh 
ℎ
 
=lim
ℎ→0
(
−𝑠𝑒𝑛𝑥(−cosℎ+1)
ℎ
+
cos𝑥∗𝑠𝑒𝑛 ℎ
ℎ
) 
= lim
ℎ→0
−𝑠𝑒𝑛 𝑥(1−cosℎ ) 
ℎ
+ lim
ℎ→0
cos𝑥 ∗ 𝑠𝑒𝑛ℎ
ℎ
 
= −𝑠𝑒𝑛 𝑥 lim
ℎ→0
(1−cosℎ ) 
ℎ
+𝑐𝑜𝑠 𝑥 lim
ℎ→0
𝑠𝑒𝑛ℎ
ℎ
 
= (−𝑠𝑒𝑛 𝑥)(0) + (cos 𝑥)(1) 
= (0) + (cos 𝑥) 
= cos 𝑥 
 
 
𝟕. 𝐥𝐢𝐦
𝒙→𝟎
√𝟏 + 𝒔𝒆𝒏𝒙 − √𝟏 − 𝒔𝒆𝒏𝒙
𝒙
 
= lim
𝑥→0
√1+𝑠𝑒𝑛𝑥−√1−𝑠𝑒𝑛𝑥
𝑥
∗
√1+𝑠𝑒𝑛𝑥+√1−𝑠𝑒𝑛𝑥
√1+𝑠𝑒𝑛𝑥+√1−𝑠𝑒𝑛𝑥
 
= lim
𝑥→0
2 𝑠𝑒𝑛 𝑥
𝑥(√1+𝑠𝑒𝑛𝑥 + √1−𝑠𝑒𝑛𝑥
 
= lim
𝑥→0
𝑠𝑒𝑛 𝑥
𝑥
∗ lim
𝑥→0
2
√1+𝑠𝑒𝑛𝑥+√1−𝑠𝑒𝑛𝑥
 
= lim
𝑥→0
2
√1+𝑠𝑒𝑛(0)+√1−𝑠𝑒𝑛(0)
→
2
√1+√1
 = 1 
 
 𝟖. 𝐥𝐢𝐦
𝒙→𝟎
√𝐜𝐨𝐬𝒙− √𝐜𝐨𝐬 𝒙
𝟑
𝒔𝒆𝒏𝟐𝒙
 
= lim
𝑥→0
1
2√cos𝑥
(−𝑠𝑒𝑛 𝑥)−
1
3 √cos𝑥
3 (−𝑠𝑒𝑛 𝑥)
2 𝑠𝑒𝑛 𝑥∗cos𝑥
 
=lim
𝑥→0
−
𝑠𝑒𝑛𝑥
2√cos𝑥
 +
𝑠𝑒𝑛𝑥
3 √cos𝑥
3 
𝑠𝑒𝑛 2𝑥
 
= lim
𝑥→0
−𝑠𝑒𝑛𝑥 ( √cos𝑥2
3
−2√cos𝑥)
6√cos𝑥 ∗ 3 √cos𝑥2
3
(2𝑠𝑖𝑛𝑥)(cos𝑥)
 
=lim
𝑥→0
−(3 √cos𝑥2
3
−2√cos𝑥)
6√cos𝑥 ∗ 3 √cos𝑥2
3
(2cos𝑥)
 
 
 
 
 
= lim
𝑥→0
−3 √cos𝑥2
3
−2√cos𝑥
6 √cos𝑥3
6
 ∗ √cos𝑥4
6
(2 cos𝑥)
 
= lim
𝑥→0
2√cos𝑥−3 √cos𝑥2
3
12 √cos𝑥7
6
(cos𝑥)
 
= lim
𝑥→0
2√cos(0)−3 √cos(0)2
3
12 √cos(0)7
6
(cos(0))
 
𝑙 =
2−3
12
 
= −
1
12
 
 
𝟗. 𝐥𝐢𝐦
𝒙→𝟎
𝐜𝐨𝐬 𝒙 − 𝐜𝐨𝐬 𝟐𝒙
𝟏 − 𝐜𝐨𝐬 𝒙 
 
= lim
𝑥→0
cos𝑥(𝑐𝑜𝑠2𝑥−𝑠𝑖𝑛2𝑥)
sin𝑥 
 
= lim
𝑥→0
cos𝑥−𝑐𝑜𝑠2𝑥+𝑠𝑖𝑛2𝑥
sin𝑥 
 
= lim
𝑥→0
cos𝑥−𝑐𝑜𝑠2𝑥
sin𝑥 
+
𝑠𝑖𝑛2𝑥
sin 𝑥
 
= lim
𝑥→0
cos𝑥(1−cos𝑥)
sin𝑥 
+
𝑠𝑖𝑛2𝑥
sin𝑥 
 
= lim
𝑥→0
(cos 𝑥 ∗
1−cos𝑥
sin𝑥
) +lim
𝑥→0
(
sin 𝑥
sin 𝑥
∗
sin𝑥
sin𝑥
) 
= 1 ∗ 1 + 2 ∗ 1 
= 2 + 1 
= 3 
 
 
𝟏𝟎. 𝐥𝐢𝐦
𝒙→𝟎
𝟏 − 𝟐 𝐜𝐨𝐬 𝒙 + 𝐜𝐨𝐬 𝟐𝒙
𝒙𝟐
 
= lim
𝑥→0
−2cos 𝑥 + 2 cos 𝑥2 − 1
𝑥2
 
= lim
𝑥→0
−2(− sin 𝑥) + 2 ∗ 2 cos 𝑥 (− sin 𝑥)
2𝑥
 
 
 
 
 
= lim
𝑥→0
−2(− sin 𝑥) + 2 sin 2𝑥
2𝑥
 
= lim
𝑥→0
sin 𝑥 − sin 2𝑥
𝑥
 
= lim
𝑥→0
cos 𝑥 − cos 2𝑥 (2)
1
 
= lim
𝑥→0
cos 𝑥 − 2 cos 2𝑥 
= lim
𝑥→0
cos(0) − 2 cos 2(0) 
= 𝑙 = 1 − 2 = −1 
 
 𝟏𝟏. 𝐥𝐢𝐦
𝒙→𝟎
𝟏−𝐜𝐨𝐬𝟕 𝒙
𝒙𝟐
 
lim
𝑥→0
1−cos7 𝑥
𝑥2
→ lim
𝑥→0
(1−cos𝑥)(1+cos𝑥+cos2 𝑥+cos2 𝑥+⋯+cos6 𝑥)
𝑥2
 
= lim
𝑥→0
(1−cos2 𝑥)(1−cos2 𝑥+cos2 𝑥 +…+cos6 𝑥)
𝑥2(1+cos2𝑥)
 
= lim
𝑥→0
sin 2𝑥
𝑥2
∗ lim
𝑥→0
1+cos2 𝑥+cos2 𝑥 +…+cos6 𝑥
1+cos2𝑥
 
= (1) (
1+1+1+1+1+1
1+1
) 
=
7
2
 
𝟏𝟐. 𝐥𝐢𝐦
𝒙→
𝒙
𝟐
𝟏 − 𝐬𝐢𝐧𝒙
(
𝝅
𝟐
− 𝒙)
𝟐 
𝐶𝑎𝑚𝑏𝑖𝑜 𝑑𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑡 =
𝜋
2
− 𝑥 ; 𝑥 =
𝜋
2
− 𝑡 
= lim
𝑡→0
1−sin(
𝜋
2
−𝑡)
(𝑡)2
→ lim
𝑡→0
1−sin(
𝜋
2
) cos 𝑡+sin 𝑡 cos(
𝜋
2
)
(𝑡)2
 
= lim
𝑡→0
1−cos 𝑡
(𝑡)2
→ lim
𝑡→0
[1−cos(𝑡)] [1+cos(𝑡)]
(𝑡)2[1+cos(𝑡)]
 
= lim
𝑡→0
1−cos2(𝑡)
(𝑡)2(1+cos𝑡)
= lim
𝑡→0
sin2(𝑡)
(𝑡)2(1+cos𝑡)
 = lim
𝑡→0
sin2(𝑡)
(𝑡)2
∗ lim
𝑡→0
1
(1+cos𝑡)
=
1
2
 
 
 
 
 
 
𝟏𝟑. 𝐥𝐢𝐦
𝒙→
𝝅
𝟒
𝐜𝐨𝐬 𝒙 − 𝐬𝐢𝐧𝒙
𝐜𝐨𝐬𝟐𝒙
 
lim
𝑥→
𝜋
4
cos𝑥−sin𝑥
cos2 𝑥−sin2 𝑥
 
= lim
𝑥→
𝜋
4
cos𝑥−sin𝑥
(cos𝑥−sin𝑥)(𝐶𝑜𝑠 𝑥+𝑠𝑒𝑛 𝑥)
 
= lim
𝑥→
𝜋
4
1
cos𝑥+sin 𝑥
 
=lim
𝑥→
𝜋
4
1
√2
2
+
√2
2
 
=
√2
2
 
 
𝟏𝟒. 𝐥𝐢𝐦
𝒙→𝟏
(𝟏 − 𝒙) 𝐭𝐚𝐧
𝝅𝒙
𝟐
 
1 − 𝑥 = ℎ ; 𝑥 = 1 − ℎ 
lim
𝑥→1
(1 + ℎ − 1) tan
𝜋(1 − ℎ)
2
 
lim
ℎ→0
 (ℎ)
sin
𝜋(1 − ℎ)
2
cos
𝜋(1 − ℎ)
2
 ; lim
ℎ→0
 (ℎ)
sin (
𝜋
2
−
𝜋ℎ
2
)
cos(
𝜋
2
−
𝜋ℎ
2
)
 ; 
lim
ℎ→0
 (ℎ)
sin
𝜋
2
∗ cos
𝜋ℎ
2
− cos
𝜋
2
∗ sin
𝜋ℎ
2
cos
𝜋
2
∗ cos
𝜋ℎ
2
+ sin
𝜋
2
∗ sin
𝜋ℎ
2
 
lim
ℎ→0
 (ℎ)
cos
𝜋ℎ
2
sin
𝜋ℎ
2
 
lim
ℎ→0
 
𝜋ℎ
2
tan 
𝜋ℎ
2
∗
𝜋
2
=
1
𝜋
2
=
2
𝜋
 
 
 
 
 
 
𝟏𝟓. 𝐥𝐢𝐦
𝒙→𝟏
𝐜𝐨𝐬 (
𝝅𝒙
𝟐
)
𝟏 − √𝒙
 
𝑥 − 1 = ℎ; 𝑥 = ℎ + 1 
lim
ℎ→0
cos (
𝜋(ℎ + 1)
2
)
1 − √ℎ + 1
 
lim
ℎ→0
cos (
𝜋ℎ
2
+
𝜋
2
)
1 − √ℎ + 1
; lim
ℎ→0
cos
𝜋ℎ
2
∗ cos
𝜋
2
− sin
𝜋ℎ
2
∗ sin
𝜋
2
1 − √ℎ + 1
; 
lim
ℎ→0
−sin
𝜋ℎ
2
1 − √ℎ + 1
∗
1 + √ℎ + 1
1 + √ℎ + 1
; lim
ℎ→0
−sin
𝜋ℎ
2
(1 + √ℎ + 1)
1 − ℎ − 1
 
𝜋
2
lim
ℎ→0
sin
𝜋ℎ
2
(1 + √ℎ + 1)
ℎ ∗
𝜋
2
 
𝜋
2
lim
ℎ→0
(
sin
𝜋ℎ
2
𝜋ℎ
2
) lim
ℎ→0
(1 + √ℎ + 1) =
𝜋
2
∗ 1 ∗ 2 = 𝜋 
 
𝟏𝟔. 𝐥𝐢𝐦
𝒙→
𝝅
𝟔
𝐬𝐢𝐧 (𝒙 −
𝝅
𝟔
)
√𝟑
𝟐
− 𝐜𝐨𝐬 𝒙
 
𝑥 −
𝜋
6
= ℎ; 𝑥 = ℎ +
𝜋
6
 
lim
ℎ→0
sin (ℎ +
𝜋
6
−
𝜋
6
)
√3
2
− cos(ℎ +
𝜋
6
)
; lim
ℎ→0
sin(ℎ)
√3
2
− cos (ℎ +
𝜋
6
)
 
lim
ℎ→0
sin(ℎ)
√3
2
− cos ℎ ∗ cos
𝜋
6
+ sin ℎ ∗ sin
𝜋
6
 
 
 
 
 
lim
ℎ→0
sin(ℎ)
√3
2
−
√3
2
cos ℎ +
1
2
sin ℎ
 
lim
ℎ→0
2 ∗ sin(ℎ)
√3(1 − cos ℎ) + sin ℎ
 
lim
ℎ→0
2 ∗ sin(ℎ)
√3(1 − cos ℎ ∗
1 + cos ℎ
1 + cos ℎ
) + sin ℎ
 
lim
ℎ→0
2 ∗ sin(ℎ)
√3(
sin2 𝑥
1 + cos ℎ
) + sin ℎ
 
lim
ℎ→0
2 ∗ sin(ℎ)
sin ℎ (
√3 sin ℎ
1 + cos ℎ
+ 1)
 
lim
ℎ→0
2
(
√3 sin ℎ
1 + cos ℎ
+ 1)
→
2
0 + 1
→ 2 
 
 
𝟏𝟕. 𝐥𝐢𝐦
𝒙→
𝝅
𝟒
𝐬𝐢𝐧 𝒙 − 𝐜𝐨𝐬 𝒙
𝟏 − 𝐭𝐚𝐧𝒙
 
lim
𝑥→
𝜋
4
sin 𝑥 − cos 𝑥
1 −
sin 𝑥
cos 𝑥
 
; lim
𝑥→
𝜋
4
(sin 𝑥 − cos 𝑥) cos 𝑥
−(−cos 𝑥 + sin 𝑥) 
lim
𝑥→
𝜋
4
− cos 𝑥 = − 
√22
 
 
𝟏𝟖. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
(
𝝅
𝟐
− 𝒙) 𝐭𝐚𝐧𝒙 
𝑥 −
𝜋
2
= ℎ; 𝑥 = ℎ +
𝜋
2
 
lim
ℎ→0
(−ℎ)
sin (ℎ +
𝜋
2
)
cos (ℎ +
𝜋
2
)
 
 
 
 
 
 
lim
ℎ→0
(−ℎ)
sin ℎ ∗ cos
𝜋
2
+ cos ℎ ∗ sin
𝜋
2
cos ℎ ∗ cos
𝜋
2
− sin ℎ ∗ sin
𝜋
2
 
lim
ℎ→0
ℎ ∗ cos ℎ
sin ℎ
; lim
ℎ→0
ℎ
tan ℎ
= 1 
 
 
𝟏𝟗. 𝐥𝐢𝐦
𝒙→𝒂
𝐬𝐢𝐧 𝒙 − 𝐬𝐢𝐧𝒂
𝒙 − 𝒂
 
lim
𝑥→𝑎
2 cos (
𝑥 + 𝑎
2
) sin (
𝑥 − 𝑎
2
)
𝑥 − 𝑎
 
ℎ = 𝑥 − 𝑎; 𝑥 = 𝑎 + ℎ 
lim
ℎ→0
2 cos (
2𝑎 + ℎ
2
) sin (
ℎ
2
)
ℎ
 
lim
ℎ→0
2
2
cos (𝑎 +
ℎ
2
) sin (
ℎ
2
)
ℎ
2
 
lim
ℎ→0
2 cos (𝑎 +
ℎ
2
) ∗ (1) = cos(𝑎 + 0) ∗ (1) = cos 𝑎 
 
 
𝟐𝟎. 𝐥𝐢𝐦
𝒙→𝒂
𝐜𝐨𝐬 𝒙 − 𝐜𝐨𝐬 𝒂
𝒙 − 𝒂
 
lim
𝑥→𝑎
−2sin (
𝑥 + 𝑎
2
) sin(
𝑥 − 𝑎
2
)
𝑥 − 𝑎
 
ℎ = 𝑥 − 𝑎; 𝑥 = 𝑎 + ℎ 
lim
ℎ→0
−2sin (
ℎ + 𝑎 + 𝑎
2
) sin(
ℎ + 𝑎 − 𝑎
2
)
ℎ + 𝑎 − 𝑎
 
 
 
 
 
lim
ℎ→0
−2/2sin (
ℎ + 2𝑎
2
) sin(
ℎ
2
)
ℎ
2
 
lim
ℎ→0
sin−(
ℎ
2
+ 𝑎) = −sin𝑎 
 
 
𝟐𝟏. 𝐥𝐢𝐦
𝒙→𝟐
𝝅
𝟑
𝐬𝐢𝐧𝟔𝒙
𝟑𝒙 − 𝟐𝝅
 
𝑥 − 2
𝜋
3
= ℎ; 𝑥 = ℎ + 2
𝜋
3
 
lim
𝑥→0
sin 6(ℎ + 2
𝜋
3
)
3(ℎ + 2
𝜋
3
) − 2𝜋
 
lim
𝑥→0
sin(6ℎ + 4𝜋)
3ℎ
; lim
𝑥→0
sin 6ℎ ∗ cos 4𝜋 + cos 6ℎ ∗ sin 4𝜋
3ℎ
 
lim
𝑥→0
2 sin 6ℎ
3ℎ ∗ 2
= 2(1) = 2 
 
𝟐𝟐. 𝐥𝐢𝐦
𝒉→𝟎
𝐬𝐢𝐧𝟐(𝒉 + 𝒂) − 𝐬𝐢𝐧𝟐 𝒂
𝒉
 
 
lim
ℎ→0
(sin(ℎ + 𝑎) − sin 𝑎)(sin(ℎ + 𝑎) + sin 𝑎)
ℎ
 
lim
ℎ→0
2 cos (
ℎ + 𝑎 + 𝑎
2
) sin(
ℎ + 𝑎 − 𝑎
2
)
ℎ
(sin(ℎ + 𝑎) + sin 𝑎) 
lim
ℎ→0
2 cos (
ℎ
2
+ 𝑎) sin(
ℎ
2
)
ℎ
2
∗ 2
(sin(ℎ + 𝑎) + sin 𝑎) 
 
 
 
 
lim
ℎ→0
2 cos (
ℎ
2
+ 𝑎)
2
(
sin (
ℎ
2
)
ℎ
2
)(sin(ℎ + 𝑎) + sin 𝑎) 
lim
ℎ→0
cos (
ℎ
2
+ 𝑎) (sin(ℎ + 𝑎) + sin 𝑎) = cos 𝑎 (sin 𝑎 + sin 𝑎) → 2 sin 𝑎 cos 𝑎 → sin 2𝑎 
 
 
𝟐𝟑. 𝐥𝐢𝐦
𝒙→𝟎
𝐬𝐢𝐧𝟑𝒙 ∗ 𝐬𝐢𝐧𝟓𝒙
(𝒙 − 𝒙𝟑)𝟐
 
lim
𝑥→0
sin 3𝑥 ∗ sin 5𝑥
(𝑥2 − 2𝑥4 + 𝑥6)
→ lim
𝑥→0
sin 3𝑥 ∗ sin 5𝑥
𝑥2(1 − 2𝑥2 + 𝑥4)
 
lim
𝑥→0
sin 3𝑥 ∗ sin 5𝑥
𝑥2(1 − 𝑥2)2
 
lim
𝑥→0
sin 3𝑥
𝑥
∗ lim
𝑥→0
sin 5𝑥
𝑥
∗ lim
𝑥→0
1
(1 − 𝑥2)2
 
lim
𝑥→0
3
sin 3𝑥
3𝑥
∗ lim
𝑥→0
5
sin 5𝑥
5𝑥
∗ lim
𝑥→0
1
(1 − 𝑥2)2
 
(3)(5) (
1
(1 − 0)2
) = 15 
 
𝟐𝟒. 𝐥𝐢𝐦
𝒙.→𝟎
𝟑 𝐬𝐢𝐧𝝅𝒙 − 𝐬𝐢𝐧 𝟑𝝅𝒙
𝒙𝟑
 
lim
𝑥.→0
3 sin 𝜋𝑥 − sin(2𝜋𝑥 + 𝜋𝑥)
𝑥3
 
lim
𝑥.→0
3 sin 𝜋𝑥 − sin 2𝜋𝑥 ∗ cos𝜋𝑥 + cos 2𝜋𝑥 ∗ sin 𝜋𝑥
𝑥3
 
lim
𝑥.→0
3 sin 𝜋𝑥 − 2 sin 𝜋𝑥 cos 𝜋𝑥 ∗ cos 𝜋𝑥 + cos 2𝜋𝑥 ∗ sin 𝜋𝑥
𝑥3
 
lim
𝑥.→0
sin 𝜋𝑥( 3 − 2 cos 𝜋𝑥 ∗ cos 𝜋𝑥 + cos 2𝜋𝑥)
𝑥3
 
lim
𝑥.→0
(
π ∗ sin 𝜋𝑥
𝜋𝑥
)
(3 − 2cos 𝜋𝑥 ∗ cos 𝜋𝑥 + cos 2𝜋𝑥)
𝑥2
 
 
 
 
 
lim
𝑥.→0
𝜋 (
3 − 3 cos2 𝜋𝑥 − sin2 𝜋𝑥
𝑥2
) 
lim
𝑥.→0
𝜋 (
3(1 − cos2 𝜋𝑥) − sin2 𝜋𝑥
𝑥2
) 
lim
𝑥.→0
𝜋3 (
2 sin2 𝜋𝑥
𝜋𝑥2
) = 2𝜋3 
 
𝟐𝟓. 𝐥𝐢𝐦
𝒙→𝟎
√𝒙𝟐 + 𝟒 − 𝟐 + 𝟐 − 𝟑 𝐜𝐨𝐬 𝒙 + 𝟏
𝟏 − 𝐜𝐨𝐬 𝒙
 
lim
𝑥→0
√𝑥2 + 4 − 2
1 − cos 𝑥
+ lim
𝑥→0
3 − 3 cos 𝑥
1 − cos 𝑥
 
lim
𝑥→0
√𝑥2 + 4 − 2
1 − cos 𝑥
∗
1 + cos 𝑥
1 + cos 𝑥
∗
√𝑥2 + 4 + 2
√𝑥2 + 4 + 2
 
lim
𝑥→0
3 − 3 cos 𝑥
1 − cos 𝑥
→ lim
𝑥→0
3(1 − cos 𝑥)
1 − cos 𝑥
→ lim
𝑥→0
3 
lim
𝑥→0
𝑥2(1 + cos 𝑥)
sin2 𝑥 (√𝑥2 + 4 + 2)
+ lim
𝑥→0
3 
lim
𝑥→0
𝑥2
sin2 𝑥
∗ lim
𝑥→0
(1 + cos 𝑥)
(√𝑥2 + 4 + 2)
+ lim
𝑥→0
3 
→ 1(
1
2
) + 3 =
7
2
 
 
𝟐𝟔. 𝐥𝐢𝐦
𝒙→
𝝅
𝟑
𝐬𝐢𝐧𝟐 𝟔𝒙 + 𝐭𝐚𝐧𝟑𝒙
𝟑𝒙 − 𝝅
 
𝑥 −
𝜋
3
= ℎ; 𝑥 = ℎ +
𝜋
3
 
lim
ℎ→0
sin2 6(ℎ +
𝜋
3
) + tan 3(ℎ +
𝜋
3
)
3(ℎ +
𝜋
3
) − 𝜋
 
 
 
lim
ℎ→0
sin2(6ℎ + 2𝜋) + tan(3ℎ + 𝜋)
3ℎ
 
lim
ℎ→0
(sin(6ℎ + 2𝜋))2 +
sin(3ℎ + 𝜋)
cos(3ℎ + 𝜋)
3ℎ
 
lim
ℎ→0
(sin 6ℎ ∗ cos 2𝜋 + cos 6ℎ ∗ sin 2𝜋)2 +
sin 3ℎ ∗ cos𝜋 + cos 3ℎ ∗ sin 𝜋
cos 3ℎ ∗ cos 𝜋 − sin 3ℎ ∗ sin 𝜋
3ℎ
 
lim
ℎ→0
(sin 6ℎ)2 +
sin 3ℎ
cos 3ℎ
3ℎ
 
lim
ℎ→0
(sin 6ℎ)2 + tan 3ℎ
3ℎ
 
lim
ℎ→0
(sin 6ℎ)2
3ℎ
+ lim
ℎ→0
tan 3ℎ
3ℎ
 
→ 0 + 1 = 1 
 
 
 
 
𝟐𝟕. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
𝐭𝐚𝐧𝟐 𝒙 (√𝟐 𝐬𝐢𝐧𝟐 𝒙 + 𝟑 𝐬𝐢𝐧𝒙 + 𝟒 − √𝐬𝐢𝐧𝟐 𝒙 + 𝟔𝐬𝐢𝐧𝒙 + 𝟐) 
lim
𝑥→
𝜋
2
tan2 𝑥 (√2 sin2 𝑥 + 3 sin 𝑥 + 4 − √sin2 𝑥 + 6 sin 𝑥 + 2) ∗
√2 sin2 𝑥 + 3 sin 𝑥 + 4 + √sin2 𝑥 + 6sin 𝑥 + 2
√2 sin2 𝑥 + 3 sin 𝑥 + 4 + √sin2 𝑥 + 6sin 𝑥 + 2
 
lim
𝑥→
𝜋
2
tan2 𝑥 (2 sin2 𝑥 + 3sin 𝑥 + 4 − sin2 𝑥 − 6 sin 𝑥 − 2)
√2 sin2 𝑥 + 3 sin 𝑥 + 4 + √sin2 𝑥 + 6 sin 𝑥 + +2
 
lim
𝑥→
𝜋
2
tan2 𝑥 (sin2 𝑥 − 3 sin 𝑥 + 2)
√2 sin2 𝑥 + 3 sin 𝑥 + 4 + √sin2 𝑥 + 6 sin 𝑥 + 2
 
lim
𝑥→
𝜋
2
sin2 𝑥 (sin 𝑥 − 2)(sin 𝑥 − 1)
(cos2 𝑥)√2 sin2 𝑥 + 3sin 𝑥 + 4 + √sin2 𝑥 + 6 sin 𝑥 + 2
 
lim
𝑥→
𝜋
2
sin2 𝑥 (sin 𝑥 − 2)(sin 𝑥 − 1)
−(sin2 𝑥 − 1)√2 sin2 𝑥 + 3 sin 𝑥 + 4 + √sin2 𝑥 + 6 sin 𝑥 + 2
 
lim
𝑥→
𝜋
2
sin2 𝑥 (sin 𝑥 − 2)
(1 + sin 𝑥)√2 sin2 𝑥 + 3 sin 𝑥 + 4 + √sin2 𝑥 + 6 sin 𝑥 + 2
 
1 ∗ (1 − 2)
(1 + 1)√2 + 3 + 4 + √1 + 6 + 2
→
1
12
 
 
 
 
 
 
 
 
𝟐𝟖. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
(𝝅 + 𝟐𝒙)𝒄𝒐𝒔 (
𝟑𝝅
𝟐
+ 𝟑𝒙)
𝒔𝒆𝒏 (
𝟑𝝅
𝟐
+ 𝟑𝒙)
 
lim
𝑥→
𝜋
2
(𝜋 + 2𝑥)𝑐𝑜𝑠 (
3𝜋
2
+ 3𝑥)
𝑠𝑒𝑛 (
3𝜋
2
+ 3𝑥)
 
Cambio de variable 
ℎ = 𝑥 +
𝜋
2
→ 𝑥 = ℎ −
𝜋
2
 
lim
ℎ→0
(𝜋 + 2ℎ − 𝑥)𝑐𝑜𝑠 (
3𝜋
2
+ 3ℎ −
3𝜋
2
)
𝑠𝑒𝑛 (
3𝜋
2
+ 3ℎ −
3𝜋
2
)
= lim
ℎ→0
2ℎ cos(3ℎ)
𝑠𝑒𝑛(3ℎ)
 
2
3
lim
ℎ→0
ℎ
𝑠𝑒𝑛(3ℎ)
= lim
ℎ→0
cos(3ℎ) =
2
3
(1) cos(0) =
2
3
 
 
 
𝟐𝟗. 𝐥𝐢𝐦
𝒙→𝟎
𝒔𝒆𝒏(𝒂 + 𝟐𝒙) − 𝟐𝒔𝒆𝒏(𝒂 + 𝒙) + 𝒔𝒆𝒏(𝒂)
𝒙𝟐
 
= lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 2𝑥) − 𝑠𝑒𝑛(𝑎 + 𝑥) − 𝑠𝑒𝑛(𝑎 + 𝑥) + 𝑠𝑒𝑛(𝑎)
𝑥2
 
= lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 2𝑥) − 2𝑠𝑒𝑛(𝑎 + 𝑥)
𝑥2
− lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 𝑥) − 𝑠𝑒𝑛(𝑎)
𝑥2
 
= lim
𝑥→0
2𝑐𝑜𝑠 (𝑎 +
3𝑥
2
) 𝑠𝑒𝑛 (
𝑥
2
)
𝑥2
− lim
𝑥→0
2𝑐𝑜𝑠 (𝑎 +
𝑥
2
) 𝑠𝑒𝑛 (
𝑥
2
)
𝑥2
 
= lim
𝑥→0
2𝑐𝑜𝑠 (𝑎 +
3𝑥
2
)
𝑥
− lim
𝑥→0
2𝑐𝑜𝑠 (𝑎 +
𝑥
2
)
𝑥
= 2 lim
𝑥→0
𝑐𝑜𝑠 (𝑎 +
3𝑥
2
)
𝑥
− lim
𝑥→0
𝑐𝑜𝑠 (𝑎 +
𝑥
2
)
𝑥
 
= 2 lim
𝑥→0
−𝑠𝑒𝑛(𝑎 + 𝑥)𝑠𝑒𝑛 (
𝑥
2
)
𝑥
= −𝟐 𝐥𝐢𝐦
𝒙→𝟎
𝑺𝒆𝒏(
𝒙
𝟐
)
𝒙
𝟐
𝐥𝐢𝐦
𝒙→𝟎
𝒔𝒆𝒏(𝒂 + 𝒙) 
= −2𝑠𝑒𝑛(𝑎 + 0) = −2𝑠𝑒𝑛(𝑎) 
 
 
 
𝟑𝟎. 𝐥𝐢𝐦
𝒙→𝟎
𝒄𝒐𝒔(𝒂 + 𝟐𝒙) − 𝟐𝒄𝒐𝒔(𝒂 + 𝒙) + 𝒄𝒐𝒔(𝒂)
𝒙𝟐
 
lim
𝑥→0
𝑐𝑜𝑠(𝑎 + 2𝑥) − 2𝑐𝑜𝑠(𝑎 + 𝑥) + cos (𝑎)
𝑥2
 
= lim
𝑥→0
𝑐𝑜𝑠(𝑎 + 2𝑥) − 𝑐𝑜𝑠(𝑎 + 𝑥) + 𝑐𝑜𝑠(𝑎) − 𝑐𝑜𝑠(𝑎 + 𝑥)
𝑥2
 
= lim
𝑥→0
𝑐𝑜𝑠(𝑎 + 2𝑥) − 𝑐𝑜𝑠(𝑎 + 𝑥)
𝑥2
− lim
𝑥→0
𝑐𝑜𝑠(𝑎 + 𝑥) − 𝑐𝑜𝑠(𝑎)
𝑥2
 
= lim
𝑥→0
−2𝑠𝑒𝑛 (𝑎 +
3𝑥
2
) 𝑠𝑒𝑛 (
𝑥
2
)
𝑥2
 
= lim
𝑥→0
𝑠𝑒𝑛 (
𝑥
2
)
𝑥
2
= lim
𝑥→0
𝑠𝑒𝑛 (𝑎 +
3𝑥
2
)
𝑥2
+ lim
𝑥→0
𝑠𝑒𝑛 (
𝑥
2
)
𝑥
2
= lim
𝑥→0
𝑠𝑒𝑛 (𝑎 +
3𝑥
2
)
𝑥
 
lim
𝑥→0
𝑠𝑒𝑛 (
𝑥
2
)
𝑥
2
= 1 
lim
𝑥→0
2𝑐𝑜𝑠(𝑎 + 𝑥)𝑠𝑒𝑛 (
𝑥
2
)
𝑥
= lim
𝑥→0
𝑠𝑒𝑛 (
𝑥
2
)
𝑥
2
∗ lim
𝑥→0
co s(𝑎 + 𝑥) 
𝑙 =−𝑐𝑜𝑠(𝑎 + 0) = −𝑐𝑜𝑠(𝑎) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
𝟑𝟏. 𝐥𝐢𝐦
𝒙→𝟎
𝒕𝒈(𝒂 + 𝟐𝒙) − 𝟐𝒕𝒈(𝒂 + 𝒙) + 𝒕𝒈(𝒂)
𝒙𝟐
 
lim
𝑥→0
𝑡𝑔(𝑎 + 2𝑥) − 𝑡𝑔(𝑎 + 𝑥) + 𝑡𝑔(𝑎) − 𝑡𝑔(𝑎 + 𝑥)
𝑥2
 
lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 2𝑥) − 𝑠𝑒𝑛(𝑎 + 𝑥)
𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)
𝑥2
− lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 𝑥) − 𝑠𝑒𝑛(𝑎)
𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)
𝑥2
 
lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥) − 𝑠𝑒𝑛(𝑎 + 𝑥)co s(𝑎 + 2𝑥)
co s(𝑎 + 2𝑥) co s(𝑎 + 𝑥)
𝑥2
− lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 𝑥)𝑐𝑜𝑠(𝑎) − 𝑠𝑒𝑛(𝑎) cos(𝑎 + 𝑥)
co s(𝑎 + 𝑥) co s(𝑎)
𝑥2
 
lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 2𝑥 − 𝑎 − 𝑥)
𝑥2𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)
− lim
𝑥→0
𝑠𝑒𝑛(𝑎 + 𝑥 − 𝑎)
𝑥2𝑐𝑜𝑠(𝑎 + 𝑥)𝑐𝑜𝑠(𝑎)
 
lim
𝑥→0
𝑠𝑒𝑛(𝑥)
𝑥2𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)
− lim
𝑥→0
𝑠𝑒𝑛(𝑥)
𝑥2𝑐𝑜𝑠(𝑎 + 𝑥)𝑐𝑜𝑠(𝑎)
 
lim
𝑥→0
𝑠𝑒𝑛(𝑥)
𝑥
lim
𝑥→0
[
1
𝑥𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)
−
1
𝑥𝑐𝑜𝑠(𝑎 + 𝑥)𝑐𝑜𝑠(𝑎)
] 
lim
𝑥→0
[
𝑐𝑜𝑠(𝑎) − 𝑐𝑜𝑠(𝑎 + 2𝑥)
𝑥𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)𝑐𝑜𝑠(𝑎)
] = lim
𝑥→0
[
−2𝑠𝑒𝑛(−𝑥)𝑠𝑒𝑛(𝑎 + 𝑥) 
𝑥𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)𝑐𝑜𝑠(𝑎)
] 
= lim
𝑥→0
[
2𝑠𝑒𝑛(𝑥)𝑠𝑒𝑛(𝑎 + 𝑥) 
𝑥𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)𝑐𝑜𝑠(𝑎)
] = lim
𝑥→0
𝑠𝑒𝑛(𝑥)
𝑥
lim
𝑥→0
[
2𝑠𝑒𝑛 (𝑎 + 𝑥)
𝑐𝑜𝑠(𝑎 + 2𝑥)𝑐𝑜𝑠(𝑎 + 𝑥)𝑐𝑜𝑠(𝑎)
] =
2𝑠𝑒𝑛(𝑎)
𝑐𝑜𝑠(𝑎)𝑐𝑜𝑠(𝑎)𝑐𝑜𝑠(𝑎)
=
2𝑠𝑒𝑛(𝑎)
𝑐𝑜𝑠3(𝑎)
 
 
 
 
 
𝟑𝟐. 𝐥𝐢𝐦𝒙→𝟎
[𝟏 − 𝐜𝐨 𝐬(𝒙)]𝟐
𝒕𝒈𝟑(𝒙) − 𝒔𝒆𝒏𝟑(𝒙) 
 
lim
𝑥→0
[1 − co s(𝑥)]2
𝑡𝑔3(𝑥) − 𝑠𝑒𝑛3(𝑥) 
= lim
𝑥→0
[1 − co s(𝑥)]2
𝑠𝑒𝑛3(𝑥)
cos3(𝑥)
− 𝑠𝑒𝑛3(𝑥) 
= lim
𝑥→0
[1 − co s(𝑥)]2
𝑠𝑒𝑛3(𝑥) [
1
cos3(𝑥)
− 1] 
= 
lim
𝑥→0
[1 − co s(𝑥)]2 cos3(𝑥)
𝑠𝑒𝑛3(𝑥)[1 − cos3(𝑥)] 
= lim
𝑥→0
[1 − co s(𝑥)]2 cos3(𝑥)
𝑠𝑒𝑛3(𝑥)[1 − 𝑐𝑜𝑠(𝑥)][1 + 𝑐𝑜𝑠(𝑥) + cos2(𝑥)] 
= 
lim
𝑥→0
[1 − co s(𝑥)]2 cos3(𝑥)
𝑠𝑒𝑛3(𝑥)𝑠𝑒𝑛(𝑥)[1 + 𝑐𝑜𝑠(𝑥) + cos2(𝑥)] 
= lim
𝑥→0
[1 − co s(𝑥)]2 cos3(𝑥)
𝑠𝑒𝑛3(𝑥)𝑠𝑒𝑛(𝑥)[1 + 𝑐𝑜𝑠(𝑥) + cos2(𝑥)] 
= 
lim
𝑥→0
[1 − co s(𝑥)] cos3(𝑥)
𝑠𝑒𝑛2(𝑥)𝑠𝑒𝑛(𝑥)[1 + 𝑐𝑜𝑠(𝑥) + cos2(𝑥)] 
= lim
𝑥→0
[1 − co s(𝑥)] cos3(𝑥)
[1 − 𝑐𝑜𝑠2(𝑥)]𝑠𝑒𝑛(𝑥)[1 + 𝑐𝑜𝑠(𝑥) + cos2(𝑥)] 
= 
lim
𝑥→0
[1 − co s(𝑥)] cos3(𝑥)
[1 − 𝑐𝑜𝑠(𝑥)][1 + 𝑐𝑜𝑠(𝑥)]𝑠𝑒𝑛(𝑥)[1 + 𝑐𝑜𝑠(𝑥) + cos2(𝑥)] 
= 
lim
𝑥→0
cos3(𝑥)
[1 + 𝑐𝑜𝑠(𝑥)]𝑠𝑒𝑛(𝑥)[1 + 𝑐𝑜𝑠(𝑥) + cos2(𝑥)] 
=
1
0
= ∞ 
 
 
 
 
 
 
 
 
 
 
 
𝟑𝟑. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
𝒄𝒐𝒔(𝒙)
 
𝝅
𝟐
− 𝒙
 
lim
𝑥→
𝜋
2
𝑐𝑜𝑠(𝑥)
 
𝜋
2
− 𝑥
 
Cambio de variable 
𝑢 = 𝑥 −
𝜋
2
→ 𝑥 = 𝑢 +
𝜋
2
 
lim
𝑥→0
𝑐𝑜𝑠(𝑢 +)
 −𝑢
= − lim
𝑥→0
𝑐𝑜𝑠(𝑢)𝑐𝑜𝑠 (
𝜋
2
) − 𝑠𝑒𝑛(𝑢)𝑠𝑒𝑛 (
𝜋
2
)
 𝑢
=
𝑠𝑒𝑛(𝑢)
𝑢
= 1 
 
 
𝟑𝟒. 𝐥𝐢𝐦
𝒙→𝟎
𝒕𝒈(𝒂𝒙) − 𝒕𝒈𝟑(𝒂𝒙)
 𝒕𝒈(𝒙)
 
lim
𝑥→0
𝑡𝑔(𝑎𝑥) − 𝑡𝑔3(𝑎𝑥)
 𝑡𝑔(𝑥)
=
(lim
𝑥→0
𝑡𝑔(𝑎𝑥)
 𝑥
− lim
𝑥→0
𝑡𝑔3(𝑎𝑥)
 𝑥
)
lim
𝑥→0
𝑡𝑔(𝑥)
 (𝑥)
= 
(𝑎 lim
𝑥→0
𝑡𝑔(𝑎𝑥)
 𝑎𝑥
− 𝑎3𝑥2 lim
𝑥→0
𝑡𝑔3(𝑎𝑥)
 (𝑎𝑥)3
)
lim
𝑥→0
𝑡𝑔(𝑥)
 (𝑥)
=
𝑎 − 0
1
= 𝑎 
 
𝟑𝟓. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
[𝟏 − 𝒔𝒆𝒏(𝒙)]𝟑
[𝟏 + 𝒄𝒐𝒔(𝟐𝒙)]𝟑 
 
lim
𝑥→
𝜋
2
[1 − 𝑠𝑒𝑛(𝑥)]3
[1 + 𝑐𝑜𝑠(𝑥) − 𝑠𝑒𝑛2(𝑥)]3 
= lim
𝑥→
𝜋
2
[1 − 𝑠𝑒𝑛(𝑥)]3
[cos2(𝑥) − cos2(𝑥)]3 
= 
lim
𝑥→
𝜋
2
[1 − 𝑠𝑒𝑛(𝑥)]3
8[cos2(𝑥)]3 
= lim
𝑥→
𝜋
2
[1 − 𝑠𝑒𝑛(𝑥)]3
8[1 − 𝑠𝑒𝑛(𝑥)]3[1 − 𝑠𝑒𝑛(𝑥)]3 
= lim
𝑥→
𝜋
2
1
8[1 − 𝑠𝑒𝑛(𝑥)]3
= 
𝑙 =
1
8(1 + 1)3
=
1
8(8)
=
1
64
 
 
 
 
 
 
𝟑𝟔. 𝐥𝐢𝐦
𝒙→𝟏
𝒔𝒆𝒏(𝟏 − 𝒙)
√𝒙 − 𝟏 
 
lim
𝑥→1
𝑠𝑒𝑛(1 − 𝑥)
 √𝑥 − 1
= lim
𝑥→1
𝑠𝑒𝑛(1 − 𝑥)(√𝑥 − 1)
 𝑥 − 1
 
Hacemos un cambio de variable 
ℎ = 1 − 𝑥 
lim
ℎ→0
(√1 − ℎ + 1)𝑠𝑒𝑛(ℎ)
 1 − ℎ − 1
= − lim
ℎ→0
(√1 − ℎ + 1)𝑠𝑒𝑛(ℎ)
 1 − ℎ − 1
= − (1 + 1) = 2 
 
𝟑𝟕. 𝐥𝐢𝐦
𝒙→𝟎
√𝒙𝟒 − 𝒙𝟒𝒔𝒆𝒏𝟐(𝒙)
𝟏 − 𝒄𝒐𝒔(𝒙)
 
lim
𝑥→0
√𝑥4 − 𝑥4𝑠𝑒𝑛2(𝑥)
1 − cos(𝑥)
= lim
𝑥→0
𝑥2√1 − 1𝑠𝑒𝑛2(𝑥)
1 − cos(𝑥)
= lim
𝑥→0
𝑥2√1 − 𝑠𝑒𝑛2(𝑥)[1 + cos(𝑥)]
[1 − cos(𝑥)][1 − cos(𝑥)]
 
lim
𝑥→0
𝑥2 cos(𝑥) [1 + cos (𝑥)]
[1 − cos2 𝑥]
= lim
𝑥→0
𝑥2 cos(𝑥) [1 + cos (𝑥)]
𝑠𝑒𝑛2(𝑥)
= lim
𝑥→0
𝑥2
𝑠𝑒𝑛2(𝑥)
= lim
𝑥→0
cos(𝑥) [1 + cos (𝑥)] 
= 1 ∗ 1 ∗ (1 + 1) = 2 
 
 
𝟑𝟖. 𝐥𝐢𝐦
𝒙→𝟎
𝟐 − √𝐜𝐨 𝐬(𝒙) − 𝐜𝐨 𝐬(𝒙)
𝒙𝟐
 
lim
𝑥→0
2 − √co s(𝑥) − 𝑐𝑜𝑠(𝑥)
𝑥2
= lim
𝑥→0
1 − √co s(𝑥) + 1 − 𝑐𝑜𝑠(𝑥)
𝑥2
 
lim
𝑥→0
[1 − √co s(𝑥)] [1 + √co s(𝑥)] 
𝑥2 [1 + √co s(𝑥)]
= lim
𝑥→0
1 − 𝑐𝑜𝑠2(𝑥) 
𝑥2 [1 + √co s(𝑥)]
+ lim
𝑥→0
1 − 𝑐𝑜𝑠2(𝑥) 
𝑥2 [1 + √co s(𝑥)]
= 
lim
𝑥→0
1 − 𝑐𝑜𝑠2(𝑥) 
𝑥2 [1 + √co s(𝑥)] [1 + √co s(𝑥)]
+ lim
𝑥→0
1𝑠𝑒𝑛2 𝑥 
𝑥2 [1 + √co s(𝑥)]
= 
 
 
 
 
lim
𝑥→0
𝑠𝑒𝑛2
𝑥2 [1 + √co s(𝑥)] [1 + √co s(𝑥)]
+ lim
𝑥→0
1 
[1 + 1]
=
1
2 ∗ 2
+
1
2
=
3
4
 
 
𝟑𝟗. 𝐥𝐢𝐦
𝒙→
𝝅
𝟑
𝟏 − 𝟐𝒄𝒐𝒔(𝒙) 
𝒔𝒆𝒏(𝒙 −
𝝅
𝟑
)
 
lim
𝑥→
𝜋
3
1 − 2𝑐𝑜𝑠(𝑥) 
𝑠𝑒𝑛 (𝑥 −
𝜋
3
)
 
Cambio de variable 
ℎ = 𝑥 −
𝜋
3
→ 𝑥 = ℎ +
𝜋
3
 
lim
ℎ→0
1 − 2𝑐𝑜𝑠 (ℎ +
𝜋
3
) 
𝑠𝑒𝑛(ℎ)
= lim
ℎ→0
1 − 2𝑐𝑜𝑠(ℎ)𝑐𝑜𝑠 (
𝜋
3
) + 2𝑠𝑒𝑛(ℎ)𝑠𝑒𝑛 (
𝜋
3
) 
𝑠𝑒𝑛(ℎ)
= 
lim
ℎ→0
1 − 𝑐𝑜𝑠(ℎ) + √3𝑠𝑒𝑛(ℎ) 
𝑠𝑒𝑛(ℎ)
= lim
ℎ→0
1 − cos2(ℎ)
1 + cos (ℎ)
+ √3𝑠𝑒𝑛(ℎ) 
𝑠𝑒𝑛(ℎ)
= 
lim
ℎ→0
[
𝑠𝑒𝑛(ℎ) 
1 + cos (ℎ)
+ √3] = √3 
 
 
 
𝟒𝟎. 𝐥𝐢𝐦
𝒙→𝟎
𝒙𝟔 
[𝒕𝒈(𝒙) − 𝒔𝒆𝒏(𝒙)]𝟐
 
lim
𝑥→0
𝑥6 
[𝑡𝑔(𝑥) − 𝑠𝑒𝑛(𝑥)]2
= lim
𝑥→0
𝑥6 
[
𝑠𝑒𝑛(𝑥)
𝑐𝑜𝑛(𝑥)
− 𝑠𝑒𝑛(𝑥)]
2 = lim𝑥→0
𝑥6 
𝑠𝑒𝑛(𝑥) [
1
𝑐𝑜𝑛(𝑥)
− 1]
2 = 
lim
𝑥→0
𝑥2
𝑠𝑒𝑛(𝑥)2
= lim
𝑥→0
𝑥4 cos2(𝑥)
[1 − cos(𝑥)]2
= lim
𝑥→0
𝑥4 cos2(𝑥)[1 + cos(𝑥)]
[1 − cos2(𝑥)]2
2
= 
lim
𝑥→0
𝑥4
𝑠𝑒𝑛2(𝑥)
= lim
𝑥→0
cos2(𝑥)[1 + cos(𝑥)]2 = (1 + 1)2 = 4 
 
 
 
 
 
 
𝟒𝟏. 𝐥𝐢𝐦
𝒙→𝟎
𝒕𝒈(𝒂𝒙)
[𝟏 − 𝒄𝒐𝒔(𝒂𝒙) + 𝒙]𝐬𝐞 𝐜(𝒂𝒙)
 
lim
𝑥→0
𝑡𝑔(𝑎𝑥)
[1 − 𝑐𝑜𝑠(𝑎𝑥) + 𝑥]𝑠𝑒𝑐(𝑎𝑥)
= lim
𝑥→0
𝑠𝑒𝑛(𝑎𝑥)
1 − cos(𝑎𝑥) + 𝑥
= lim
𝑥→0
𝑠𝑒𝑛(𝑎𝑥)
(1 − cos2(𝑎𝑥))
1 + cos (𝑎𝑥)
+ 𝑥
= 
lim
𝑥→0
𝑠𝑒𝑛(𝑎𝑥)
𝑠𝑒𝑛2(𝑎𝑥)
1 + cos (𝑎𝑥)
+ 𝑥
= lim
𝑥→0
1
𝑠𝑒𝑛2(𝑎𝑥)
1 + cos (𝑎𝑥)
+
𝑎𝑥
𝑎 𝑠𝑒𝑛 (𝑎𝑥)
 
𝑙 =
1
0 +
1
𝑎
= 𝑎 
 
𝟒𝟐. 𝐥𝐢𝐦
𝒙→𝟓
𝐬𝐢𝐧(𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓)
𝒙𝟑 + 𝟓𝒙𝟐 − 𝟏𝟐𝟓𝒙 + 𝟑𝟕𝟓
 
lim
𝑥→5
sin[(𝑥 − 5)(𝑥 − 5)]
(𝑥 − 5)(𝑥 − 5)(𝑥 + 15)
 
lim
𝑥→5
sin[(𝑥 − 5)(𝑥 − 5)]
(𝑥 − 5)(𝑥 − 5)
(
1
(𝑥 + 15)
) 
(
1
(5 + 15)
) 
1
20
 
 
 
𝟒𝟑. 𝐥𝐢𝐦
𝒙→𝟎
𝟐
𝐬𝐢𝐧𝟐 𝒙
−
𝟏
𝟏 − 𝐜𝐨𝐬 𝒙
 
lim
𝑥→0
2
1 − cos2 𝑥
−
1
1 − cos 𝑥
 
lim
𝑥→0
2
(1 − cos 𝑥)(1 + cos 𝑥)
−
1
1 − cos 𝑥
 
 
 
 
lim
𝑥→0
2 − 1 − cos 𝑥
(1 − cos 𝑥)(1 + cos 𝑥)
 
lim
𝑥→0
1 − cos 𝑥
(1 − cos 𝑥)(1 + cos 𝑥)
 
lim
𝑥→0
1
(1 + cos 𝑥)
 
1
2
 
 
 
𝟒𝟒. 𝐥𝐢𝐦
𝒙→𝟎
𝐱𝐬𝐢𝐧(𝐬𝐢𝐧𝟐𝒙)
𝟏 − 𝐜𝐨𝐬(𝐬𝐢𝐧𝟒𝒙)
 
lim
𝑥→0
xsin(sin 2𝑥)
1 − cos(sin 4𝑥)
(
sin 2𝑥
sin 2𝑥
) 
lim
𝑥→0
xsin(sin 2𝑥)
sin 2𝑥
(
sin 2𝑥
1 − cos(sin 4𝑥)
) 
lim
𝑥→0
𝑥 (
sin 2𝑥
1 − cos(sin 4𝑥)
) (
2𝑥
2𝑥
) 
lim
𝑥→0
𝑥 (
sin 2𝑥
2𝑥
) (
2𝑥
1 − cos(sin 4𝑥)
) 
lim
𝑥→0
(𝑥) (
2𝑥
1 − cos(sin 4𝑥)
) 
lim
𝑥→0
(
2𝑥2
1 − cos(sin 4𝑥)
) (
1 + cos(sin 4𝑥)
1 + cos(sin 4𝑥)
) 
lim
𝑥→0
(
2𝑥2
1 − cos(sin 4𝑥)
) (
1 + cos(sin 4𝑥)
1 + cos(sin 4𝑥)
) 
lim
𝑥→0
(
2𝑥2(1 + cos(sin 4𝑥))
1 − cos2(sin 4𝑥)
) 
lim
𝑥→0
2𝑥2[1 + cos(sin 4𝑥)]
sin2(sin 4𝑥)
 
 
 
 
lim
𝑥→0
2𝑥2[1 + cos(sin 4𝑥)]
1
[
1
sin2(sin 4𝑥)
] 
lim
𝑥→0
2𝑥2[1 + cos(sin 4𝑥)]
1
[
1
sin(sin 4𝑥)
(
sin 4𝑥
sin 4𝑥
)]
2
 
lim
𝑥→0
2𝑥2[1 + cos(sin 4𝑥)]
1
[
1
sin4𝑥
(
4𝑥
4𝑥
)]
2
 
lim
𝑥→0
2𝑥2[1 + cos(sin 4𝑥)]
1
(
1
4𝑥
)
2
 
lim
𝑥→0
2𝑥2[1 + cos(sin 4𝑥)]
16𝑥2
 
lim
𝑥→0
[1 + cos(sin 4𝑥)]
8
 
[1 + 1]
8
=
2
8
=
1
4
 
 
𝟒𝟓. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
𝐜𝐨𝐬(𝒙)
√𝟏 − 𝐬𝐢𝐧(𝒙)
 
lim
𝑥→
𝜋
2
cos(𝑥)
√1 − sin(𝑥)
(
√1 + sin(𝑥)
√1 + sin(𝑥)
) 
lim
𝑥→
𝜋
2
cos(𝑥) (√1 + sin(𝑥))
√1 − sin2(𝑥)
 
lim
𝑥→
𝜋
2
cos(𝑥) (√1 + sin(𝑥))
√cos2(𝑥)
 
lim
𝑥→
𝜋
2
cos(𝑥) (√1 + sin(𝑥))
cos(𝑥)
 
lim
𝑥→
𝜋
2
(√1 + sin(𝑥))
1
 
√1 + 1 = √2 
 
 
 
 
 
𝟒𝟔. 𝐥𝐢𝐦
𝒙→𝟎
𝐜𝐨𝐬(𝒙) − √𝐜𝐨𝐬(𝟐𝒙)
𝒙 𝐬𝐢𝐧(𝒙)
 
lim
𝑥→0
cos(𝑥) − √cos(2𝑥)
𝑥 sin(𝑥)
(
cos(𝑥) + √cos(2𝑥)
cos(𝑥) + √cos(2𝑥)
) 
lim
𝑥→0
cos2(𝑥) − cos(2𝑥)
𝑥 sin(𝑥) (cos(𝑥) + √cos(2𝑥))
 
lim
𝑥→0
1 − sen2(𝑥) − [1 − 2 sen2(𝑥)]
𝑥 sin(𝑥) (cos(𝑥) + √cos(2𝑥))
 
lim
𝑥→0
sen2(𝑥)
𝑥 sin(𝑥) (cos(𝑥) + √cos(2𝑥))
 
lim
𝑥→0
sen(𝑥)
𝑥
(
1
(cos(𝑥) + √cos(2𝑥))
) 
1
(cos(0) + √cos(0))
=
1
2
 
 
𝟒𝟕. 𝐥𝐢𝐦
𝒙→𝝅
𝟏 − 𝐬𝐢𝐧 (
𝒙
𝟐
)
𝐜𝐨𝐬 (
𝒙
𝟐
) [𝐜𝐨𝐬 (
𝒙
𝟒
) − 𝐬𝐢𝐧 (
𝒙
𝟒
)]
 
lim
𝑥→𝜋
1 − sin (
𝑥
2
)
cos (
𝑥
2
) [cos (
𝑥
4
) − sin (
𝑥
4
)]
[
1 + sin (
𝑥
2
)
1 + sin (
𝑥
2
)
] [
cos (
𝑥
4
) + sin (
𝑥
4
)
cos (
𝑥
4
) + sin (
𝑥
4
)
] 
lim
𝑥→𝜋
1 − sin2 (
𝑥
2
) [cos (
𝑥
4
) + sin (
𝑥
4
)]
cos (
𝑥
2
) [cos2 (
𝑥
4
) − sin2 (
𝑥
4
)] [1 + sin (
𝑥
2
)]
 
lim
𝑥→𝜋
cos2 (
𝑥
2
) [cos (
𝑥
4
) + sin (
𝑥
4
)]
cos (
𝑥
2
) [cos2 (
𝑥
4
) − sin2 (
𝑥
4
)] [1 + sin (
𝑥
2
)]
 
 
 
 
 
lim
𝑥→𝜋
𝑐𝑜𝑠 (
𝑥
2
) [cos (
𝑥
4
) + sin (
𝑥
4
)]
[cos (
2𝑥
4
)] [1 + sin (
𝑥
2
)]
 
lim
𝑥→𝜋
𝑐𝑜𝑠 (
𝑥
2
) [cos (
𝑥
4
) + sin (
𝑥
4
)]
cos (
𝑥
2
) [1 + sin (
𝑥
2
)]
 
lim
𝑥→𝜋
[cos (
𝑥
4
) + sin (
𝑥
4
)]
[1 + sin (
𝑥
2
)]
 
√2
2
+
√2
2
1 + 1
=
√2
2
 
 
𝟒𝟖. 𝐥𝐢𝐦
𝒙→𝒂
𝐜𝐨𝐬(𝒂 + 𝒙) − 𝐜𝐨𝐬(𝒂 − 𝒙)
𝒙
 
Aplicando la identidad trigonométrica “producto de senos”lim
𝑥→𝑎
−2sin(𝑎) sin(𝑥)
𝑥
 
−2sin(𝑎) 
 
𝟒𝟗. 𝐥𝐢𝐦
𝒙→𝟎
√𝟏 + 𝐬𝐢𝐧(𝒙) − √𝟏 − 𝐬𝐢𝐧(𝒙)
𝐭𝐚𝐧(𝒙)
 
lim
𝑥→0
√1 + sin(𝑥) − √1 − sin(𝑥)
tan(𝑥)
(
√1 + sin(𝑥) + √1 − sin(𝑥)
√1 + sin(𝑥) + √1 − sin(𝑥)
) 
lim
𝑥→0
1 + sin(𝑥) − 1 + sin(𝑥)
tan(𝑥) [√1 + sin(𝑥) + √1 − sin(𝑥)]
 
lim
𝑥→0
2 sin(𝑥)
tan(𝑥) [√1 + sin(𝑥) + √1 − sin(𝑥)]
 
 
 
 
lim
𝑥→0
2 sin(𝑥)
sin(𝑥)
cos(𝑥)
(
1
[√1 + sin(𝑥) + √1 − sin(𝑥)]
) 
lim
𝑥→0
2 cos(𝑥)
1
(
1
[√1 + sin(𝑥) + √1 − sin(𝑥)]
) 
2 (
1
2
) = 1 
 
𝟓𝟎. 𝐥𝐢𝐦
𝒙→𝟎
𝟏 − 𝐜𝐨𝐬 𝒙√𝐜𝐨𝐬(𝟐𝒙)
𝒙𝟐
 
lim
𝑥→0
1 − cos 𝑥 √cos(2𝑥)
𝑥2
(
1 + cos 𝑥 √cos(2𝑥)
1 + cos 𝑥 √cos(2𝑥)
) 
lim
𝑥→0
1 − cos2(𝑥) cos(2𝑥)
𝑥2 (1 + cos 𝑥 √cos(2𝑥))
 
lim
𝑥→0
1 − cos2(𝑥) [1 − 2 sin2(𝑥)]
𝑥2 (1 + cos 𝑥 √cos(2𝑥))
 
lim
𝑥→0
1 − cos2(𝑥) + 2 sin2(𝑥) cos2(𝑥)
𝑥2 (1 + cos 𝑥 √cos(2𝑥))
 
lim
𝑥→0
sin2(𝑥) + 2 sin2(𝑥) cos2(𝑥)
𝑥2 (1 + cos 𝑥 √cos(2𝑥))
 
lim
𝑥→0
sin2(𝑥) [1 + 2 cos2(𝑥)]
𝑥2 (1 + cos 𝑥 √cos(2𝑥))
 
lim
𝑥→0
[1 + 2 cos2(𝑥)]
(1 + cos 𝑥 √cos(2𝑥))
[
sin(𝑥)
𝑥
]
2
 
1 + 2(1)
1 + 1
=
3
2
 
 
 
 
 
𝟓𝟏. 𝐥𝐢𝐦
𝒙→𝟎
𝟒𝐜𝐨𝐬 𝒙 −𝐜𝐨𝐬(𝟐𝒙) − 𝟑
𝒙 𝐬𝐢𝐧𝟑(𝒙)
 
lim
𝑥→0
4 cos 𝑥 −cos(2𝑥) − 3 + 4 − 4
𝑥 sin3(𝑥)
 
lim
𝑥→0
4 cos 𝑥 − 4
𝑥 sin3(𝑥)
+
−cos(2𝑥) − 3 + 4
𝑥 sin3(𝑥)
 
lim
𝑥→0
(4)
cos 𝑥 − 1
𝑥 sin3(𝑥)
+
−cos(2𝑥) + 1
𝑥 sin3(𝑥)
 
lim
𝑥→0
(−4)
1 − cos 𝑥
𝑥 sin3(𝑥)
−
cos(2𝑥) − 1
𝑥 sin3(𝑥)
(
cos(2𝑥) + 1
cos(2𝑥) + 1
) 
lim
𝑥→0
(−4)
1 − cos 𝑥
𝑥 sin3(𝑥)
(
1 + cos 𝑥
1 + cos 𝑥
) −
cos2(2𝑥) − 1
𝑥 sin3(𝑥)
 
lim
𝑥→0
(−4)
1 − cos2 𝑥
𝑥 sin3(𝑥) (1 + cos 𝑥)
−
cos2(2𝑥) − 1
𝑥 sin3(𝑥) (cos(2𝑥) + 1)
 
lim
𝑥→0
(−4)
sen2 𝑥
𝑥 sin3(𝑥) (1 + cos 𝑥)
+
sen2(2𝑥)
𝑥 sin3(𝑥) (cos(2𝑥) + 1)
 
lim
𝑥→0
(−4)
1
𝑥 sin(𝑥) (1 + cos 𝑥)
+
4sin2(𝑥) cos2(𝑥)
𝑥 sin3(𝑥) (cos(2𝑥) + 1)
 
lim
𝑥→0
−4
𝑥 sin(𝑥) (1 + cos 𝑥)
+
4 cos2(𝑥)
𝑥 sin(𝑥) (cos(2𝑥) + 1)
 
lim
𝑥→0
(−4)(
1
𝑥 sin(𝑥) (1 + cos 𝑥)
−
cos2(𝑥)
𝑥 sin(𝑥) (cos(2𝑥) + 1)
) 
lim
𝑥→0
(−4)(
(cos(2𝑥) + 1) − (1 + cos 𝑥)(cos2(𝑥))
𝑥 sin(𝑥) (1 + cos 𝑥)(cos(2𝑥) + 1)
) 
lim
𝑥→0
(−4)(
cos(2𝑥) + 1 − cos3(𝑥) − cos2(𝑥)
𝑥 sin(𝑥) (1 + cos 𝑥)(cos(2𝑥) + 1)
) 
1 + 1 − 1 − 1
0
= ∞ 
 
 
 
 
 
 
𝟓𝟐. 𝐥𝐢𝐦
𝒙→
𝝅
𝟑
𝟏 − 𝟒 𝐜𝐨𝐬𝟐(𝒙)
𝟖 𝐬𝐢𝐧 (𝒙 −
𝝅
𝟑
)
 
𝑧 = 𝑥 −
𝜋
3
 
𝑥 = 𝑧 +
𝜋
3
 
lim
𝑥→
𝜋
3
1 − 4 cos2 (𝑧 +
𝜋
3
)
8 sin(𝑧)
 
lim
𝑧→0
1 − [cos(𝑧) − √3 sin(𝑧)]
2
8 sin(𝑧)
 
lim
𝑧→0
1 − cos2(𝑧) + 2√3 sin(𝑧) cos(𝑧) − 3 sin2(𝑧)
8 sin(𝑧)
 
lim
𝑧→0
sin 2(𝑧) + 2√3 sin(𝑧) cos(𝑧) − 3 sin2(𝑧)
8 sin(𝑧)
 
lim
𝑧→0
sin(𝑧) [sin(𝑧) + 2√3cos(𝑧) − 3 sin(𝑧)]
8 sin(𝑧)
 
lim
𝑧→0
[sin(𝑧) + 2√3cos(𝑧) − 3 sin(𝑧)]
8
 
[2√3]
8
=
√3
4
 
 
𝟓𝟑. 𝐥𝐢𝐦
𝒙→𝟏
(𝒙 − 𝟏) 𝐬𝐢𝐧 (
𝟏
𝒙 − 𝟏
) 
lim
𝑥→1
sin (
1
𝑥 − 1
)
1
𝑥 − 1
= 1 
 
 
 
 
 
 
𝟓𝟒. 𝐥𝐢𝐦
𝒙→𝟏
𝟐𝒙𝟑 − 𝐜𝐨𝐬(𝒙 − 𝟏) − 𝟏
𝒙𝟐 − 𝟏
 
lim
𝑥→1
2𝑥3 + 2 − cos(𝑥 − 1) − 1 − 2
𝑥2 − 1
 
lim
𝑥→1
2𝑥3 − 2
𝑥2 − 1
+
− cos(𝑥 − 1) + 1
𝑥2 − 1
 
lim
𝑥→1
2(𝑥3 − 1)
𝑥2 − 1
+
1 − cos(𝑥 − 1)
𝑥2 − 1
 
lim
𝑥→1
2(𝑥 − 1)(𝑥2 + 2𝑥 + 1)
(𝑥 − 1)(𝑥 + 1)
+
1 − cos(𝑥 − 1)
(𝑥 − 1)(𝑥 + 1)
 
2(12 + 2 + 1)
(1 + 1)
= 3 
 
𝟓𝟓. 𝐥𝐢𝐦
𝒙→𝟏
𝟓 𝐬𝐢𝐧(𝒙) − 𝟑 𝐜𝐨𝐬(𝒙) + 𝟑
𝟐 𝐭𝐚𝐧(𝒙) + 𝟏 − 𝐜𝐨𝐬(𝒙)
 
lim
𝑥→1
5 sin(𝑥) − 3 cos(𝑥) + 3
2 sen(𝑥)
cos(𝑥)
+ 1 − cos(𝑥)
 
lim
𝑥→1
5 sin(𝑥) − 3 cos(𝑥) + 3
2 sen(𝑥) + cos(𝑥) − cos2(𝑥)
cos(𝑥)
 
lim
𝑥→1
5 sin(𝑥) cos(𝑥) − 3 cos2(𝑥) + 3 cos(𝑥)
2 sen(𝑥) + cos(𝑥) − cos2(𝑥)
 
lim
𝑥→1
5 sin(𝑥) cos(𝑥) − 3 cos2(𝑥) + 3 cos(𝑥)
𝑥
2 sen(𝑥) + cos(𝑥) − cos2(𝑥)
𝑥
 
lim
𝑥→1
5 sin(𝑥) cos(𝑥)
𝑥
+
3 cos(𝑥) (1 − cos(𝑥))
𝑥
2 sen(𝑥)
𝑥
+
cos(𝑥) (1 − cos(𝑥))
𝑥
 
5
2
 
 
 
 
 
56. 𝐥𝐢𝐦
𝒙→𝓪
𝒔𝒆𝒏𝟐𝒙−𝒔𝒆𝒏𝟐𝓪
𝒙−𝓪
 
lim
𝑥→𝒶
(𝑠𝑒𝑛 𝑥 − 𝑠𝑒𝑛 𝒶)(𝑠𝑒𝑛 𝑥 + 𝑠𝑒𝑛 𝒶)
𝑥 − 𝒶
 
ℎ = 𝑥 − 𝒶 ⇒ 𝑥 = ℎ + 𝒶 
lim
ℎ→0
(𝑠𝑒𝑛 (ℎ + 𝑎) − 𝑠𝑒𝑛 𝒶)
ℎ
lim
ℎ→0
 𝑠𝑒𝑛(ℎ + 𝒶) + 𝑠𝑒𝑛 𝒶 
Identidad trigonométrica 
lim
ℎ→0
2𝑐𝑜𝑠 (𝒶 +
ℎ
2
) 𝑠𝑒𝑛 (
ℎ
2
)
ℎ
 [𝑠𝑒𝑛(𝒶) + 𝑠𝑒𝑛 𝒶] 
2𝑠𝑒𝑛(𝒶) lim
ℎ→0
𝑠𝑒𝑛 (
ℎ
2
)
ℎ
2
lim
ℎ→0
 𝑐𝑜𝑠 (𝒶 +
ℎ
2
) 
2 𝑠𝑒𝑛 (𝒶)(1) 𝑐𝑜𝑠(𝒶 + 0) = 2 𝑠𝑒𝑛(𝒶) 𝑐𝑜𝑠(𝒶) = 𝑆𝑒𝑛(2𝒶) 
 
𝟓𝟕. 𝐥𝐢𝐦
𝒙→𝟎
𝒄𝒐𝒔 𝒙 − 𝒄𝒐𝒔(𝒔𝒆𝒏 𝟐𝒙)
𝒙𝟐
 
Identidad trigonométrica 
lim
𝑥→0
−2 𝑠𝑒𝑛 [
𝑥 + 𝑠𝑒𝑛(2𝑥)
2
] 𝑠𝑒𝑛 [
𝑥 + 𝑠𝑒𝑛(2𝑥)
2
] 
𝑥2
 
−2lim
𝑥→0
𝑠𝑒𝑛 [
𝑥 + 𝑠𝑒𝑛(2𝑥)
2
]
[
𝑥 + 𝑠𝑒𝑛(2𝑥)
2
]
[
𝑥 + 𝑠𝑒𝑛(2𝑥)
2
] lim
𝑥→0
𝑠𝑒𝑛 [
𝑥 + 𝑠𝑒𝑛(2𝑥)
2
] 
𝑥2
 
 −2lim
𝑥→0
[
𝑥 + 𝑠𝑒𝑛(2𝑥)
2
] lim
𝑥→0
𝑠𝑒𝑛 [
𝑥 − 𝑠𝑒𝑛(2𝑥)
2
] 
𝑥
 
−2lim
𝑥→0
[
1
2
+
 𝑠𝑒𝑛(2𝑥)
2𝑥
] lim
𝑥→0
𝑠𝑒𝑛 [
𝑥 − 𝑠𝑒𝑛(2𝑥)
2
] 
[
𝑥 − 𝑠𝑒𝑛(2𝑥)
2
] 
[
𝑥 − 𝑠𝑒𝑛(2𝑥)
2
] 
 
 
 
−2(
1
2
+ 1) lim
𝑥→0
𝑠𝑒𝑛 [
𝑥 − 𝑠𝑒𝑛(2𝑥)
2
] 
[
𝑥 − 𝑠𝑒𝑛(2𝑥)
2
] 
lim
𝑥→0
[
𝑥 − 𝑠𝑒𝑛(2𝑥)
2
] 
−3(1)lim
𝑥→0
[
𝑥 − 𝑠𝑒𝑛(2𝑥)
2
] = −3 (
1
2
− 1) = 
3
2
 
 
 
𝟓𝟖. 𝐥𝐢𝐦
𝒙→𝟎
√
𝒄𝒐𝒔 𝒎𝒙 − 𝒄𝒐𝒔 𝒏𝒙
𝒙𝟐
𝟑
 
lim
𝑥→0
√
(1 − 𝑐𝑜𝑠 𝑚𝑥) − (1 − 𝑐𝑜𝑠 𝑛𝑥)
𝑥2
3
 = lim
𝑥→0
√
1 − 𝑐𝑜𝑠 𝑛𝑥
𝑥2
3
 − lim
𝑥→0
√
1 − 𝑐𝑜𝑠 𝑚𝑥
𝑥2
3
 
lim
𝑥→0
√
𝑠𝑒𝑛2𝑛𝑥
𝑥2(1 + 𝑐𝑜𝑠 𝑛𝑥)
3
 − lim
𝑥→0
√
𝑠𝑒𝑛2𝑚𝑥
𝑥2(1 + 𝑐𝑜𝑠 𝑚𝑥)
3
 
lim
𝑥→0
√(
𝑛 𝑠𝑒𝑛 𝑛𝑥
𝑛𝑥
)
2 1
1 + 𝑐𝑜𝑠 𝑛𝑥
3
 − lim
𝑥→0
√(
𝑚 𝑠𝑒𝑛 𝑚𝑥
𝑚𝑥
)
2 1
1 + 𝑐𝑜𝑠 𝑚𝑥
3
 
√
𝑛2
2
3
− √
𝑚2
2
3
 = √
𝑛2 − 𝑚2
2
3
 
 
 
 
 
 
 
 
 
 
 
 
 𝟓𝟗. 𝐥𝐢𝐦
𝒙→𝟎
𝟏−𝒄𝒐𝒔 𝒙 𝒄𝒐𝒔 𝟐𝒙 𝒄𝒐𝒔 𝟑𝒙
𝟏−𝒄𝒐𝒔 𝒙
 
lim
𝑥→0
1 − 𝑐𝑜𝑠 (𝑥) 𝑐𝑜𝑠 (2𝑥) 𝑐𝑜𝑠 (3𝑥) + 𝑐𝑜𝑠(2𝑥) 𝑐𝑜𝑠 (3𝑥) − 𝑐𝑜𝑠(2𝑥) 𝑐𝑜𝑠(3𝑥)
1 − 𝑐𝑜𝑠 𝑥
 
lim
𝑥→0
1 + 𝑐𝑜𝑠 (2𝑥) 𝑐𝑜𝑠 (3𝑥)[1 − 𝑐𝑜𝑠(𝑥)] − 𝑐𝑜𝑠(2𝑥) 𝑐𝑜𝑠(3𝑥)
1 − 𝑐𝑜𝑠 𝑥
 
lim
𝑥→0
 𝑐𝑜𝑠 (2𝑥) 𝑐𝑜𝑠 (3𝑥)[1 − 𝑐𝑜𝑠(𝑥)]
1 − 𝑐𝑜𝑠 𝑥
+ lim
𝑥→0
1 − 𝑐𝑜𝑠(2𝑥) 𝑐𝑜𝑠(3𝑥)
1 − 𝑐𝑜𝑠 𝑥
 
lim
𝑥→0
 𝑐𝑜𝑠 (2𝑥) 𝑐𝑜𝑠 (3𝑥) + lim
𝑥→0
1 − 𝑐𝑜𝑠(2𝑥) 𝑐𝑜𝑠(3𝑥) + 𝑐𝑜𝑠 (3𝑥) − 𝑐𝑜𝑠(3𝑥)
1 − 𝑐𝑜𝑠 𝑥
 
𝑐𝑜𝑠 (0) 𝑐𝑜𝑠 (0) + lim
𝑥→0
1 + 𝑐𝑜𝑠(3𝑥)[1 − 𝑐𝑜𝑠(𝑥)] − 𝑐𝑜𝑠(3𝑥)
1 − 𝑐𝑜𝑠 𝑥
 
1 + lim
𝑥→0
𝑐𝑜𝑠(3𝑥)[1 − 𝑐𝑜𝑠(𝑥)]
1 − 𝑐𝑜𝑠 𝑥
 + lim
𝑥→0
1 − 𝑐𝑜𝑠(3𝑥)
1 − 𝑐𝑜𝑠 𝑥
 
1 + lim
𝑥→0
𝑐𝑜𝑠(3𝑥)[1 − 𝑐𝑜𝑠2(𝑥) + 𝑠𝑒𝑛2(𝑥)]
1 − 𝑐𝑜𝑠 𝑥
 + lim
𝑥→0
[1 − 𝑐𝑜𝑠2(3𝑥)][1 − 𝑐𝑜𝑠 𝑥]
[1 − 𝑐𝑜𝑠2(𝑥)][1 − 𝑐𝑜𝑠(3𝑥)]
 
1 + lim
𝑥→0
𝑐𝑜𝑠(3𝑥)[ 𝑠𝑒𝑛2(𝑥) + 𝑠𝑒𝑛2(𝑥)]
1 − 𝑐𝑜𝑠 𝑥
 + lim
𝑥→0
[𝑠𝑒𝑛2(3𝑥)][1 + 𝑐𝑜𝑠 𝑥]
[𝑠𝑒𝑛2(𝑥)][1 + 𝑐𝑜𝑠(3𝑥)]
 
1 + lim
𝑥→0
2𝑐𝑜𝑠(3𝑥)[ 1 − 𝑐𝑜𝑠2(𝑥)]
1 − 𝑐𝑜𝑠 𝑥
 + lim
𝑥→0
[
𝑠𝑒𝑛2(3𝑥)
𝑥2
] [1 + 𝑐𝑜𝑠 𝑥]
[
𝑠𝑒𝑛2(𝑥)
𝑥2
] [1 + 𝑐𝑜𝑠(3𝑥)]
 
1 + lim
𝑥→0
2𝑐𝑜𝑠(3𝑥)[ 1 − 𝑐𝑜𝑠(𝑥)][ 1 + 𝑐𝑜𝑠(𝑥)]
1 − 𝑐𝑜𝑠 𝑥
 + 9lim
𝑥→0
𝑠𝑒𝑛2(3𝑥)
(3𝑥)2
lim
𝑥→0
𝑥2
𝑠𝑒𝑛2(𝑥)
lim
𝑥→0
[1 + 𝑐𝑜𝑠 𝑥]
[1 + 𝑐𝑜𝑠(3𝑥)]
 
1 + lim
𝑥→0
2𝑐𝑜𝑠(3𝑥)[ 1 − 𝑐𝑜𝑠(𝑥)] + 9
1 + 1
1 + 1
 = 1 + 2(1 + 1) + 9 = 14 
 
 
 
 
 
 
𝟔𝟎. 𝐥𝐢𝐦
𝒙→𝟎
𝒔𝒆𝒏(𝟐𝒙 + 𝓪) − 𝟐𝒔𝒆𝒏( 𝒙 + 𝓪) + 𝒔𝒆𝒏 𝓪
𝒙𝟐
 
lim
𝑥→0
𝑠𝑒𝑛(2𝑥 + 𝒶) − 𝑠𝑒𝑛( 𝑥 + 𝒶) 
𝑥2
+ lim
𝑥→0
𝑠𝑒𝑛(𝒶) − 𝑠𝑒𝑛( 𝑥 + 𝒶) 
𝑥2
 
Identidad Trigonométrica 
lim
𝑥→0
2𝑐𝑜𝑠(
3𝑥
2
+ 𝒶)𝑠𝑒𝑛( 
𝑥
2
) 
𝑥2
+ lim
𝑥→0
2𝑐𝑜𝑠(
𝑥
2
+ 𝒶)𝑠𝑒𝑛(− 
𝑥
2
) 
𝑥2
 
lim
𝑥→0
𝑠𝑒𝑛( 
𝑥
2
) 
𝑥
2
 lim
𝑥→0
𝑐𝑜𝑠(
3𝑥
2
+ 𝒶) 
𝑥
 − lim
𝑥→0
𝑠𝑒𝑛(
𝑥
2
) 
𝑥
2
 lim
𝑥→0
𝑐𝑜𝑠(
𝑥
2
+ 𝒶 
𝑥
 
lim
𝑥→0
𝑐𝑜𝑠 (
3𝑥
2
+ 𝒶) 
𝑥
 − lim
𝑥→0
𝑐𝑜𝑠 (
𝑥
2
+ 𝒶) 
𝑥
 − lim
𝑥→0
𝑐𝑜𝑠 (
3𝑥
2
+ 𝒶) − 𝑐𝑜𝑠 (
𝑥
2
+ 𝒶) 
𝑥
 
lim
𝑥→0
−2𝑠𝑒𝑛(𝑥 + 𝒶)𝑠𝑒𝑛 (
𝑥
2
) 
𝑥
 = − lim
𝑥→0
𝑠𝑒𝑛 (
𝑥
2
) 
𝑥
2
 lim
𝑥→0
 𝑠𝑒𝑛(𝑥 + 𝒶) 
−(1)𝑠𝑒𝑛(0 + 𝒶) = −𝑠𝑒𝑛(𝒶) 
 
𝟔𝟏. 𝐥𝐢𝐦
𝒙→𝟏
(
𝒙𝟐 − 𝒙
𝒔𝒆𝒏(𝒙 − 𝟏)
+
√𝒙 − 𝟏
𝒔𝒆𝒏(𝒙 −𝟏)
) 
lim
𝑥→1
(
𝑥2 − 𝑥
𝑠𝑒𝑛(𝑥 − 1)
) + lim
𝑥→1
(
√𝑥 − 1
𝑠𝑒𝑛(𝑥 − 1)
) 
1 +
1
2
=
3
2
 
 
𝟔𝟐. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
𝟏 − 𝒔𝒆𝒏 𝒙 
𝒄𝒐𝒔 𝒙
 
lim
𝑥→
𝝅
𝟐
−𝑐𝑜𝑠 𝑥
−𝑠𝑒𝑛 𝑥
 = lim
𝑥→
𝝅
𝟐
 𝑐𝑜𝑡 𝑥 = 𝑐𝑜𝑡 (
𝜋
2
) 
 
 
 
 
 
𝟔𝟑. 𝐥𝐢𝐦
𝒙→
𝝅
𝟑
𝟏 − 𝟒𝒄𝒐𝒔𝟐 𝒙 
𝟖𝒔𝒆𝒏(𝒙 −
𝝅
𝟑
)
 
1
8
lim
𝑥→
𝜋
3
1 − 4𝑐𝑜𝑠2 𝑥 
𝑠𝑒𝑛(𝑥 −
𝜋
3
)
 
1
8
lim
𝑥→
𝜋
3
4𝑠𝑖𝑛( 2𝑥) 
1
2
𝑐𝑜𝑠(𝑥) +
√3
2
𝑠𝑒𝑛(𝑥)
 = 
1
8
lim
𝑥→
𝜋
3
4𝑠𝑖𝑛( 2𝑥) 
𝑐𝑜𝑠(𝑥) + √3 𝑠𝑒𝑛(𝑥)
 
1
8
∗ 
8𝑠𝑖𝑛( 2 ∗
𝜋
3
) 
𝑐𝑜𝑠(
𝜋
3
) + √3 𝑠𝑒𝑛(
𝜋
3
)
 = 
√3
4
 
 
 
𝟔𝟒. 𝐥𝐢𝐦
𝒙→𝟎
𝟐(𝟏 − 𝒄𝒐𝒔𝒙) 
𝒙 𝒕𝒈𝒙
 
lim
𝑥→0
2(1 − 𝑐𝑜𝑠𝑥) 
𝑥 𝑡𝑔𝑥
∗
(1 + cos 𝑥)
(1 + cos 𝑥)
 
lim
𝑥→0
2(1 − cos2 𝑥) 
𝑥 𝑡𝑔𝑥(1 + cos 𝑥)
 
2 lim
𝑥→0
(sin2 𝑥) 
𝑥 𝑡𝑔𝑥(1 + cos 𝑥)
→ 2 lim
𝑥→0
sin 𝑥
 𝑡𝑔𝑥(1 + cos 𝑥)
 
2 lim
𝑥→0
sin 𝑥
sin 𝑥
cos 𝑥
(1 + cos 𝑥)
→ 2 lim
𝑥→0
cos 𝑥
 (1 + cos 𝑥)
→ 2 (
1
1 + 1
) = 1 
 
 
 
 
 
 
 
 
 
 
 
𝟔𝟓. 𝐥𝐢𝐦
𝒙→𝟎
𝒕𝒈 (𝒙 +
𝝅
𝟒
) − 𝟏 
𝒔𝒆𝒏 𝒙
 
lim
𝑥→0
2
−𝑠𝑖𝑛(2𝑥) + 1
 
𝑐𝑜𝑠 𝑥
 
lim
𝑥→0
2
𝑐𝑜𝑠(𝑥)(−𝑠𝑖𝑛(2𝑥) + 1)
 = 
2
𝑐𝑜𝑠(0)(−𝑠𝑖𝑛(2(0)) + 1)
 = 2 
 
 
𝟔𝟔. 𝐥𝐢𝐦
𝒙→𝟎
√𝟏 + 𝒙
𝟑
− 𝒄𝒕𝒈 𝒙 − 𝟏 + 𝒄𝒐𝒔 𝒆𝒄𝒙
𝒙 
 
lim
𝑥→0
√1 + 𝑥
3
− 1
𝑥 
 lim
𝑥→0
𝑐𝑜𝑠𝑒𝑐 (𝑥) − 𝑐𝑡𝑔 𝑥
𝑥 
 
lim
𝑥→0
√1 + 𝑥
3
− 1
𝑥 
 
√1 + 𝑥
3
+ 1
√1 + 𝑥
3
+ 1
 = lim
𝑥→0
1 + 𝑥 + 1
𝑥√(1 + 𝑥)2
3
+ 2√1 + 𝑥
3
+ 12
 
lim
𝑥→0
1
(√(1 + 𝑥)2
3
+ 2√1 + 𝑥
3
+ 1)
 =
1
(1 + 1 + 1)
 = 
1
3
 
lim
𝑥→0
𝑐𝑜𝑠𝑒𝑐 (𝑥) − 𝑐𝑡𝑔 𝑥
𝑥 
 =
1
2
 
1
3
∗
1
2
= 
1
6
 
 
𝟔𝟕. 𝐥𝐢𝐦
𝒙→𝟎
𝟏 − 𝒄𝒐𝒔𝒙 
𝒔𝒆𝒏𝟐𝒙
 
lim
𝑥→0
1 − 𝑐𝑜𝑠𝑥 
𝑠𝑒𝑛2𝑥
= lim
𝑥→0
1 
1 + 𝑐𝑜𝑠𝑥
 
1 
1 + 𝑐𝑜𝑠(0)
=
1 
2
 
 
 
 
 
𝟔𝟖. 𝐥𝐢𝐦
𝒙→𝟎
(𝟐 − 𝟐 𝒄𝒐𝒔 𝒙)𝟐
𝒙𝟒
 
lim
𝑥→0
4𝑠𝑖𝑛(𝑥)(2 − 2 𝑐𝑜𝑠 𝑥)
4𝑥3
 
lim
𝑥→0
𝑠𝑖𝑛(𝑥)(−2 𝑐𝑜𝑠 (𝑥) + 2)
𝑥3
 
lim
𝑥→0
𝑐𝑜𝑠(𝑥)(−2 𝑐𝑜𝑠 𝑥 + 2) + 2𝑠𝑒𝑛2(𝑥)
3𝑥2
 = lim
𝑥→0
(−2 𝑠𝑖𝑛(𝑥) + 4𝑠𝑖𝑛(2 𝑥)
6𝑥
 
lim
𝑥→0
(− sin(𝑥) + 2 sin(2 𝑥))
3𝑥
 = lim
𝑥→0
(− cos(𝑥) + 4 cos(2 𝑥))
3
 
(−𝑐𝑜𝑠(0) + 4𝑐𝑜𝑠(2 (0))
3
 = 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
LÍMITES DE LA 
FORMA 𝒇(𝒙)𝒈(𝒙) 
 
 
 
 
 
 
 
 
 
 
𝟏. 𝐥𝐢𝐦
𝒙→∞
(
𝒙𝟑 + 𝟐𝒙 + 𝟑
𝒙𝟑 + 𝟒
)
𝒙𝟐+𝟐
 
lim
𝑥→∞
(1 +
𝑥3 + 2𝑥 + 3
𝑥3 + 4
 −)
𝑥2+2
= lim
𝑥→∞
(1 +
𝑥3 + 2𝑥 − 3 − 𝑥3 − 4
𝑥3 + 4
 )
𝑥2+2
 
lim
𝑥→∞
{(1 +
2𝑥 − 1
𝑥3 + 4
 )
𝑥2+2
}
(𝑥2+2)(
2𝑥−1
𝑥2+1
)
 = 𝑒
lim 
𝑥→∞
(1+2/𝑥2)(2−𝑥)
144𝑥3 = 𝑒1(2) = 𝑒2 
 
𝟐. 𝐥𝐢𝐦
𝒙→∞
(
𝒙𝟐 − 𝟐𝒙 + 𝟏
𝒙𝟐 − 𝟒𝒙 + 𝟐
)
𝒙
 
lim
𝑥→∞
(1 +
𝑥2 − 2𝑥 + 1
𝑥2 − 4𝑥 + 2
− 1)
𝑥
 = lim
𝑥→∞
(1 +
𝑥2 − 2𝑥 + 1 − 𝑥2 − 4𝑥 + 2
𝑥2 − 4𝑥 + 2
)
𝑥
 
lim
𝑥→∞
(1 +
2𝑥 − 1
𝑥2 − 4𝑥 + 2
)
𝑥
 = lim
𝑥→∞
{(1 +
2𝑥 − 1
𝑥3 + 4
)
(
𝑥3+4
2𝑥−1)
}
𝑥(
2𝑥−1
𝑥3+4
)
 
𝑒
lim 
𝑥→∞
(1+2/𝑥2)(2−𝑥)
144𝑥3 = 𝑒1(2) = 𝑒2 
 
𝟑. 𝐥𝐢𝐦
𝒙→∞
(
𝟑𝒙 − 𝟒
𝟑𝒙 + 𝟐
)
𝒙+𝟏
𝟑
 
lim
𝑥→∞
(1 +
3𝑥 − 4
3𝑥 + 2
− 1)
𝑥+1
3
 
= lim
𝑥→∞
(1 +
3𝑥 − 4 − 3𝑥 − 2
3𝑥 + 2
)
𝑥+4
3
lim
𝑥→∞
{(1 +
−6
3𝑥 + 2
)
(
3𝑥+2
−6 )
}
3+1
3 (
−6
3𝑥+2)
 = 𝑒
lim
𝑥→∞
−2
−2𝑥 
= 𝑒−2/3 
 
𝟒. 𝐥𝐢𝐦
𝒙→𝟎
(𝐜𝐨 𝐬(𝒙) + 𝒔𝒆𝒏(𝒙))
𝟏
𝒙 
 
lim
𝑥→0
(𝑐𝑜𝑠(𝑥) )
1
𝑥 lim
𝑥→0
(1 + 𝑡𝑔(𝑥))
1
𝑥 
lim
𝑥→0
(1 − 1 + 𝑐𝑜𝑠(𝑥) )
1
𝑥 lim
𝑥→0
{(1 + 𝑡𝑔(𝑥))𝑐𝑜𝑡(𝑥)}
𝑡𝑔(𝑥)
𝑥 
lim
𝑥→0
(1 − [1 + 𝑐𝑜𝑠(𝑥)] )
1
𝑥 𝑒 = lim
𝑥→0
(1 − 2 𝑠𝑒𝑛2(
𝑥
2
) )
1
𝑥
 𝑒 
lim
𝑥→0
(1 − 2 𝑠𝑒𝑛2(
𝑥
2
) )
1
𝑥𝑠𝑒𝑛
2(
𝑥
2)
 𝑒 = 𝑒−1(0)𝑒 = 𝑒 
 
 
 
 
 
 
 
 
 
𝟓. 𝐥𝐢𝐦
𝒙→𝟎
𝐥𝐧 (𝓪 + 𝒙) − 𝒍𝒏𝓪
𝒙
 
 
lim
𝑥→0
1
𝑥
𝑙𝑛 (
𝒶 + 𝑥
𝒶
) = lim
𝑥→0
𝒶
𝑥
(
1
𝒶
) 𝑙𝑛 (1 +
𝑥
𝒶
) 
1
𝒶
lim
𝑥→0
𝑙𝑛 (1 +
𝑥
𝒶
)
𝒶
𝑥
 = 
1
𝒶
 𝑙𝑛(𝑒) = 
1
𝒶
 
 
 
𝟔. 𝐥𝐢𝐦
𝒙→∞
𝒙(𝒍𝒏(𝒙 + 𝓪) − 𝒍𝒏 𝒙) 
lim
𝑥→∞
𝑥 𝑙𝑛 (
𝑥 + 𝒶
𝑥
) = lim
𝑥→∞
𝑥
𝒶
(𝒶) 𝑙𝑛 (
𝑥 + 𝒶
𝑥
)
𝑥
𝒶
 = 𝒶 𝑙𝑛(𝑒) = 𝒶 
 
𝟕. 𝐥𝐢𝐦
𝒙→𝟎
(
𝒙𝟐 − 𝟐𝒙 + 𝟑
𝒙𝟐 − 𝟑𝒙 + 𝟐
)
𝒔𝒆𝒏(𝒙)
𝒙
 
= (
3
2
)
lim
𝑥→𝑜
𝑠𝑒𝑛(𝑥)
𝑥
= 
3
2
 
 
𝟖. 𝐥𝐢𝐦
𝒙→∞
(
𝒙𝟐 − 𝟏
𝒙𝟐 + 𝟏
)
𝒙−𝟏
𝒙+𝟏
 
lim
𝑥→∞
(1 +
𝑥2 − 1
𝑥2 + 1
− 1)
𝑥−1
𝑥+1
 = lim
𝑥→∞
(1 +
𝑥2 − 1 − 𝑥2 − 1
𝑥2 + 1
)
𝑥−1
𝑥+1
 
lim
𝑥→∞
(1 +
2
𝑥2 + 1
)
𝑥−1
𝑥+1
 lim
𝑥→∞
{(1 +
2
𝑥2 + 1
)
𝑥2+1
2
}
2
𝑥2+1
∗
𝑥−1
𝑥+1 
 
𝑒
lim
𝑥→∞
2
𝑥2+1
∗
𝑥−1
𝑥+1 = 𝑒0(1) = 𝑒0 = 1 
 
𝟗. 𝐥𝐢𝐦
𝒙→∞
(
𝒙𝟐 + 𝟏
𝒙𝟐 − 𝟐
)
𝒙𝟐
 
lim
𝑥→∞
(1 +
𝑥2 + 1
𝑥2 − 2
− 1)
𝑥2
 = lim
𝑥→∞
(1 +
𝑥2 + 1 − 𝑥2 + 2
𝑥2 − 2
)
𝑥2
 
 
 
 
 
 
 
lim
𝑥→∞
{(1 +
3
𝑥2 − 2
)
𝑥2−2
3
}
3𝑥2
𝑥2−2
 = 𝑒
lim
𝑥→∞
3𝑥2
𝑥2−2 = 𝑒
lim
𝑥→∞
3
1−
2
𝑥2 = 𝑒
3
1−0 = 𝑒3 
 
𝟏𝟎. 𝐥𝐢𝐦
𝒙→𝒐
√𝟏 − 𝟐𝒙
𝒙
 
lim
𝑥→𝑜
(1 − 2𝑥)
1
𝑥 = lim
𝑥→𝑜
{(1 − 2𝑥)
1
−2𝑥}
−2
 = 𝑒−2 
 
 
𝟏𝟏. 𝐥𝐢𝐦
𝒙→∞
(
𝒙 + 𝓪
𝒙 − 𝓪
)
𝒙
 
lim
𝑥→∞
(1 +
𝑥 + 𝒶
𝑥 − 𝒶
− 1)
𝑥
 = lim
𝑥→∞
[(1 +
2𝒶
𝑥 − 𝒶
)
𝑥−𝒶
2𝒶
]
2𝒶
𝑥−𝒶∗𝑥
 
𝑒
lim
𝑥→∞
2𝒶
1−
𝒶
𝑥 = 𝑒2𝒶 
 
12. 𝐥𝐢𝐦
𝒙→∞
(
𝒙𝟑+𝟐𝒙+𝟑
𝒙𝟑+𝟒
)
𝟏−𝒙𝟑
𝒙
 
lim
𝑥→∞
(1 +
𝑥3 + 2𝑥 + 3
𝑥3 + 4
− 1)
1−𝑥3
𝑥
 = lim
𝑥→∞
(1 +
𝑥3 + 2𝑥 + 3 − 𝑥3 − 4
𝑥3 + 4
)
1−𝑥3
𝑥
 
lim
𝑥→∞
{(1 +
2𝑥 − 1
𝑥3 + 4
)
𝑥3+4
2𝑥−1
}
(2𝑥−1)(1−𝑥3)
(𝑥3+4)𝑥
 = 𝑒
lim
𝑥→∞
(2𝑥−1)(1−𝑥3)
(𝑥3+4)𝑥 
𝑒
lim
𝑥→∞
(2−
1
𝑥)(
1
𝑥3
−1)
(1+
4
𝑥3
)
 = 𝑒2(−1) = 𝑒−2 
 
𝟏𝟑. 𝐥𝐢𝐦
𝒙→𝟎
[𝟏 − 𝐬𝐢𝐧(𝟑𝒙)]
𝟏
𝟐𝒙 
lim
𝑥→0
[1 − sin(3𝑥)]
1
2𝑥(
−sin(3𝑥)
−sin(3𝑥)
)
 
lim
𝑥→0
[1 − sin(3𝑥)]
−
1
sin(3𝑥)
(
−sin(3𝑥)
2𝑥 ) 
[𝑒]
lim
𝑥→0
 (
−sin(3𝑥)
2𝑥 ) 
𝑒−
3
2 
 
 
 
 
 
 
𝟏𝟒. 𝐥𝐢𝐦
𝒙→𝟎
[𝒆𝒙 + 𝒙]
𝒎
𝒙 
lim
𝑥→0
[𝑒𝑥 (
𝑒𝑥 + 𝑥
𝑒𝑥
)]
𝑚
𝑥
 
lim
𝑥→0
[1 +
𝑥
𝑒𝑥
]
𝑚
𝑥
[𝑒𝑥]
𝑚
𝑥 
lim
𝑥→0
[1 +
𝑥
𝑒𝑥
]
𝑚
𝑥 (
𝑒𝑥
𝑒𝑥
)
[𝑒𝑥]
𝑚
𝑥 
lim
𝑥→0
[1 +
𝑥
𝑒𝑥
]
𝑒𝑥
𝑥 (
𝑚
𝑒𝑥
)
[𝑒𝑥]
𝑚
𝑥 
[𝑒𝑥]
(
𝑚
𝑒0
)
𝑒𝑚 
𝑒2𝑚 
 
𝟏𝟓. 𝐥𝐢𝐦
𝒙→𝟎
[𝒆𝒙 + 𝒙]
𝟏
𝒙 
lim
𝑥→0
[𝑒𝑥 (
𝑒𝑥 + 𝑥
𝑒𝑥
)]
1
𝑥
 
lim
𝑥→0
𝑒 [(1 +
𝑥
𝑒𝑥
)]
1
𝑥
 
lim
𝑥→0
𝑒 [(1 +
𝑥
𝑒𝑥
)]
1
𝑥(
𝑒𝑥
𝑒𝑥
)
 
lim
𝑥→0
𝑒 [(1 +
𝑥
𝑒𝑥
)]
𝑒𝑥
𝑥 (
1
𝑒𝑥
)
 
𝑒[𝑒]
(
1
𝑒𝑥
)
= 𝑒2 
 
𝟏𝟔. 𝐥𝐢𝐦
𝒙→𝟎
(𝒙 + 𝒂)𝒙+𝒂(𝒙 + 𝒃)𝒙+𝒃
(𝒙 + 𝒂 + 𝒃)𝟐𝒙+𝒂+𝒃
 
lim
𝑥→0
(𝑥 + 𝑎)𝑥+𝑎(𝑥 + 𝑏)𝑥+𝑏
(𝑥 + 𝑎 + 𝑏)𝑥+𝑎(𝑥 + 𝑎 + 𝑏)𝑥+𝑏
 
lim
𝑥→0
(
𝑥 + 𝑎
𝑥 + 𝑎 + 𝑏
)
𝑥+𝑎
(
𝑥 + 𝑏
𝑥 + 𝑎 + 𝑏
)
𝑥+𝑎
 
lim
𝑥→0
(
𝑥 + 𝑎
𝑥 + 𝑎 + 𝑏
+ 1 − 1)
𝑥+𝑎
(
𝑥 + 𝑏
𝑥 + 𝑎 + 𝑏
+ 1 − 1)
𝑥+𝑎
 
 
 
 
 
lim
𝑥→0
(
𝑥 + 𝑎 − 𝑥 − 𝑎 − 𝑏
𝑥 + 𝑎 + 𝑏
+ 1)
𝑥+𝑎
(
𝑥 + 𝑏 − 𝑥 − 𝑎 − 𝑏
𝑥 + 𝑎 + 𝑏
+ 1)
𝑥+𝑎
 
lim
𝑥→0
(−
𝑏
𝑥 + 𝑎 + 𝑏
+ 1)
𝑥+𝑎
(−
𝑎
𝑥 + 𝑎 + 𝑏
+ 1)
𝑥+𝑎
 
lim
𝑥→0
(−
𝑏
𝑥 + 𝑎 + 𝑏
+ 1)
𝑥+𝑎
(−
𝑎
𝑥 + 𝑎 + 𝑏
+ 1)
𝑥+𝑎
 
lim
𝑥→0
(−
𝑏
𝑥 + 𝑎 + 𝑏
+ 1)
(
𝑥+𝑎+𝑏
𝑏 )(
−𝑏(𝑥+𝑎)
𝑥+𝑎+𝑏 ) 
(−
𝑎
𝑥 + 𝑎 + 𝑏
+ 1)
(
𝑥+𝑎+𝑏
𝑎 )(
−𝑎(𝑥+𝑏)
𝑥+𝑎+𝑏 )
 
(𝑒)
(
−𝑏(0+𝑎)
0+𝑎+𝑏 ) (𝑒)
(
−𝑎(0+𝑏)
0+𝑎+𝑏 ) = (𝑒)−(𝑎+𝑏) 
 
𝟏𝟕. 𝐥𝐢𝐦
𝒙→𝟎
( √𝟏 − 𝐬𝐢𝐧(√𝟑)
√𝟑
)
𝟏
𝐬𝐢𝐧(√𝟑)
 
lim
𝑥→0
[1 − sin(√3)]
(
1
√3
)(
1
sin(√3)
)
 
𝑒
(
1
√3
)
 
 
𝟏𝟖. 𝐥𝐢𝐦
𝒙→∞
(
 
𝟒
√𝟏𝟔𝒙 𝐬𝐢𝐧 (
𝟏
𝟒𝒙
))
 
𝒙𝐬𝐢𝐧
𝟏
𝟑𝒙
 
1
4𝑥
= 0 
1
4𝑥
= ℎ 
1
4ℎ
= 𝑥 
= lim
ℎ→0
(
 
4
√4
ℎ
sin(ℎ))
 
1
4ℎ sin
4ℎ
3
= lim
ℎ→0
(
 
4
√4
ℎ
sin(ℎ))
 
sin
4ℎ
3
4ℎ
= √2
3
 
 
 
 
 
 
 
 
 
 
𝟏𝟗. 𝐥𝐢𝐦
𝒙→𝟎
𝐥𝐧(𝒙 + 𝒉) − 𝐥𝐧(𝒙)
𝒙
 
lim
𝑥→0
ln(𝑥 + ℎ) − ln(𝑥)
𝑥
 
lim
𝑥→0
ln (
𝑥 + ℎ
𝑥
)
𝑥
 
lim
𝑥→0
1
𝑥
ln (
𝑥 + ℎ
𝑥
) 
lim
𝑥→0
1
𝑥
ln (1 +
ℎ
𝑥
)
𝑥
ℎ
 =
1
𝑥
 
 
𝟐𝟎.𝐥𝐢𝐦
𝒙→𝟎
(
𝟏 + 𝐭𝐚𝐧𝒙
𝟏 − 𝐭𝐚𝐧𝒙
)
𝟏
𝐬𝐢𝐧(𝒙)
 
lim
𝑥→0
(
1 + tan𝑥
1 − tan𝑥
+ 1 − 1)
1
sin(𝑥)
 
lim
𝑥→0
(
1 + tan𝑥 − 1 + tan 𝑥
1 − tan𝑥
+ 1)
1
sin(𝑥)
 
lim
𝑥→0
(
2 tan𝑥
1 − tan𝑥
+ 1)
(
1−tan 𝑥
2 tan𝑥 )
1
sin(𝑥)
(
2 tan𝑥
1−tan𝑥)
 
(𝑒)
1
sin(𝑥)
(
2 tan𝑥
1−tan𝑥
)
 = (𝑒)2 
 
𝟐𝟏. 𝐥𝐢𝐦
𝒙→𝟎
(
𝟏 + 𝐭𝐚𝐧𝒙
𝟏 − 𝐬𝐢𝐧 𝒙
)
𝟏
𝐬𝐢𝐧(𝒙)
 
lim
𝑥→0
(
1 + tan𝑥
1 − sin 𝑥
− 1 + 1)
1
sin(𝑥)
 
lim
𝑥→0
(
tan 𝑥 + sin 𝑥
1 − sin 𝑥
+ 1)
1
sin(𝑥)
 
lim
𝑥→0
(
tan 𝑥 + sin 𝑥
1 − sin 𝑥
+ 1)
1
sin(𝑥)
(
1−sin𝑥
tan𝑥+sin𝑥
)(
tan𝑥+sin𝑥
1−sin𝑥
)
 
(𝑒)
1
sin(𝑥)
(
tan𝑥+sin𝑥
1−sin𝑥
)
 
(𝑒)2 
 
 
 
 
 
 
𝟐𝟐. 𝐥𝐢𝐦
𝒙→𝟎
(√𝟐 − √𝐜𝐨𝐬(𝒙))
𝟏
𝒙𝟐
 
lim
𝑥→0
(2 − √cos(𝑥))
(
1
2)
1
𝑥2 
lim
𝑥→0
(2 −
cos(𝑥)
√cos(𝑥)
)
(
1
2)
1
𝑥2
 
lim
𝑥→0
(1 −
√cos(𝑥) + cos(𝑥)
√cos(𝑥)
)
(
1
2)
1
𝑥2
(−
√cos(𝑥)+cos(𝑥)
√cos(𝑥)
)(−
√cos(𝑥)
√cos(𝑥)+cos(𝑥)
)
 
lim
𝑥→0
(𝑒)0 
(𝑒)0 = 1 
 
𝟐𝟑. 𝐥𝐢𝐦
𝒙→𝟎
(𝐜𝐨𝐬(𝒙))
𝟏
𝒙𝟐 
lim
𝑥→0
(1 − 1 + cos(𝑥))
1
𝑥2 
lim
𝑥→0
(1 + 2sin2
𝑥
2
)
(
1
2sin2
)
1
𝑥2
(
2sin2
1 )
 
lim
𝑥→0
(1 + 2sin2
𝑥
2
)
(
1
2sin2
𝑥
2
)
1
𝑥2
(
2sin2
𝑥
2
 
1 )
 
(𝑒)
1
𝑥2
(
2sin2
𝑥
2 
1 )
 
(𝑒)
1
2 
 
𝟐𝟒. 𝐥𝐢𝐦
𝒙→𝟎
[𝐜𝐨𝐬(√
𝟓𝒂
𝒙
)]
𝟏
𝒃𝒙
 
lim
𝑥→0
[1 − 1 + cos(√
5𝑎
𝑥
)]
1
𝑏𝑥
 
 
 
 
lim
𝑥→0
[1 + 2sin2 (
1
2
√
5𝑎
𝑥
)]
1
𝑏𝑥
 
lim
𝑥→0
[1 + 2sin2 (
1
2
√
5𝑎
𝑥
)](
 
 1
2sin2(
1
2
√5𝑎
𝑥 )
)
 
 
2sin2(
1
2
√5𝑎
𝑥 )
𝑏𝑥
 
[𝑒]
2 sin2(
1
2
√5𝑎
𝑥 )
𝑏𝑥 
[𝑒][
 
 
 
 sin2(
1
2
√5𝑎
𝑥
)
𝑏𝑥
]
 
 
 
 
2
 
[𝑒][
 
 
 
 sin (
1
2
√5𝑎
𝑥 )
√𝑏𝑥
(
 
1
2
√
5𝑎
𝑥
1
2
√5𝑎
𝑥 )
 
]
 
 
 
 
2
2
 
[𝑒][
 
 
 1
2
√5𝑎
𝑥
√𝑏𝑥
]
 
 
 
2
2
 
[𝑒]
1
2 
 
𝟐𝟓. 𝐥𝐢𝐦
𝒙→𝟎
[
(𝒆𝒙 + 𝒙)𝒕𝒈(𝒙)
(𝟏 + 𝒔𝒆𝒏 (𝒙))
𝒙]
𝒄𝒕𝒈(𝒙)
𝐱 
 
lim
𝑥→0
[
(𝑒𝑥+𝑥)𝑡𝑔(𝑥)
(1+𝑠𝑒𝑛 (𝑥))
𝑥]
𝑐𝑡𝑔(𝑥)
x
 
 = lim
𝑥→0
(𝑒𝑥 +𝑥)
1
𝑥
(1+𝑠𝑒𝑛(𝑥))
𝑐𝑡𝑔(𝑥) = lim𝑥→0
[𝑒𝑥 (1+
𝑥
𝑒𝑥
)]
1
𝑥
(1+𝑠𝑒𝑛(𝑥))
𝑐𝑜𝑠(𝑥)
𝑠𝑒𝑛(𝑥)
 
=e lim
𝑥→0
[(1+
𝑥
𝑒𝑥
)
𝑒𝑥
𝑥 ]
𝑥
𝑒𝑥
+
1
𝑥
[(1+𝑠𝑒𝑛(𝑥))
1
𝑠𝑒𝑛(𝑥)]
𝑐𝑜𝑠(𝑥) = 𝐞.
𝒆
𝒆
=𝒆 
 
 
 
 
 
 
 
 
𝟐𝟔. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
[𝒔𝒆𝒏(𝒙)]𝒕𝒈(𝒙) 
𝑙 = lim
𝑥→
𝜋
2
 [𝑠𝑒𝑛(𝑥)]𝑡𝑔(𝑥) Cambio de variable ℎ = 𝑥 −
𝜋
2
 ⇒ 𝑥 = ℎ + 
𝜋
2
 
𝑙 = lim
ℎ→0
[𝑠𝑒𝑛 (ℎ +
𝜋
2
)]
𝑡𝑔(ℎ+
𝜋
2
)= lim
ℎ→0
[𝑐𝑜𝑠 (ℎ)]𝑐𝑡𝑔 (ℎ) 
 
 
𝑙 = lim
ℎ→0
[1 − 1 + 𝑐𝑜𝑠(ℎ)]𝑐𝑡𝑔(𝑥) = 𝐿 = lim
ℎ→0
[1 − (1 + 𝑐𝑜𝑠(ℎ))]
𝑐𝑡𝑔(ℎ)
 
Pero 1 − 𝑐𝑜𝑠 (𝑥) = 2𝑠𝑒𝑛2 ( 
𝑥
2
) 
𝑙 = lim
ℎ→0
[1 − 2𝑠𝑒𝑛2 (
ℎ
2
)]
𝑐𝑡𝑔(𝑏)
 = 𝑒
−lim
ℎ→0
 
𝑠𝑒𝑛 
ℎ
2
ℎ
2
 lim
ℎ→0
𝑠𝑒𝑛(ℎ+2)𝑐𝑜𝑠(ℎ)
𝑠𝑒𝑛(ℎ)
 = 𝑒
−lim
ℎ→0
 
𝑠𝑒𝑛(
ℎ
2)
ℎ
2
 lim
ℎ→0
(
ℎ
2)𝑐𝑜𝑠 
(ℎ) 
𝑠𝑒𝑛(ℎ)
 
 
𝑙 = 𝑒
−lim
ℎ→0
 
ℎ
𝑠𝑒𝑛 (ℎ)
 lim
ℎ→0
𝑐𝑜𝑠 (ℎ)
2 = 𝑒
1
2 
 
 
𝟐𝟕. 𝐥𝐢𝐦
𝒙→𝟎
(
𝒄𝒐𝒔(𝒙)
𝒄𝒐𝒔(𝟐𝒙)
)
𝟏
𝒙𝟐
 
 
𝒍 = lim
𝑥→0
(
𝑐𝑜𝑠(𝑥)
𝑐𝑜𝑠(2𝑥)
)
1
𝑥2 
lim
𝑥→0
(1 +
𝑐𝑜𝑠(𝑥)
𝑐𝑜𝑠(2𝑥)
− 1)
1
𝑥2
 
𝐿 = lim
𝑥→0
(1 +
𝑐𝑜𝑠(𝑥) − 𝐶𝑂𝑆(2𝑋)
𝑐𝑜𝑠(2𝑥)
)
1
𝑥2
 
𝑙 = lim 
𝑥→0
{(1 +
𝐶𝑂𝑆(𝑋) − 𝐶𝑂𝑆 (2𝑋)
𝐶𝑂𝑆(2𝑋)
)
𝐶𝑂𝑆(2𝑋)
[𝐶𝑂𝑆 (𝑋)−𝐶𝑂𝑆(2𝑋)]
}
[𝐶𝑂𝑆(𝑋)−𝐶𝑂𝑆(2𝑋)]
𝑋2𝐶𝑂𝑆 (2𝑋)
 
𝑙 = 𝑒
lim 
𝑥→∞
[𝑐𝑜𝑠(𝑥)−𝑐𝑜𝑠2(𝑥)+𝑠𝑒𝑛2(𝑥)]
𝑥2 𝑐𝑜𝑠(2𝑥) = 𝑒
lim 
𝑥→∞
𝑠𝑒𝑛2
𝑥 lim 𝑥→∞
 
[
𝑐𝑜𝑠(𝑥)
1+𝑐𝑜𝑠 (𝑥)]
𝑐𝑜𝑠(2𝑥) = 𝑒 
3
2 
 
 
 
 
 
 
 
 
 
𝟐𝟖. 𝐥𝐢𝐦
𝒙→𝟎
(𝟏 + 𝒙𝟐)
𝒄𝒕𝒈𝟐(𝒙)
 
lim
𝑥→0
(1 + 𝑥2)𝑐𝑡𝑔
2(𝑥) = lim
𝑥→0
[(1 + 𝑥2)
1
𝑥2]
𝑥2𝑐𝑡𝑔2(𝑥)
 
lim
𝑥→0
𝑒
𝑥2
𝑠𝑒𝑛2(𝑥)
cos2(𝑥)
= 𝑒 
 
𝟐𝟗. 𝐥𝐢𝐦
𝒙→𝟎
(𝟏 + 𝒕𝒈√𝒙)
𝟏
𝟐√𝒙 
lim
𝑥→0
(1 + 𝑡𝑔√𝑥)
1
2√𝑥 = lim
𝑥→0
[(1 + 𝑡𝑔√𝑥)
1
𝑡𝑔√𝑥]
𝑡𝑔√𝑥
2√𝑥
= 𝑒
1
2 = √𝑒 
 
𝟑𝟎. 𝐥𝐢𝐦
𝒙→𝟎
(𝒄𝒐𝒔(𝒙))
𝟏
𝒔𝒆𝒏(𝒙) 
lim
𝑥→0
(𝑐𝑜𝑠(𝑥))
1
𝑠𝑒𝑛(𝑥) = lim
𝑥→0
(1 − 1 + 𝑐𝑜𝑠(𝑥))
1
𝑠𝑒𝑛(𝑥) 
lim
𝑥→0
[1 − (1 + 𝑐𝑜𝑠(𝑥))]
1
𝑠𝑒𝑛(𝑥) 
1 − 𝑐𝑜𝑠(𝑥) = 2𝑠𝑒𝑛2 (
𝑥
2
) 
lim
𝑥→0
[1 − 2𝑠𝑒𝑛2 (
𝑥
2
) ]
 
1
𝑠𝑒𝑛(𝑥)
= lim
𝑥→0
{lim
𝑥→0
[1 − 2𝑠𝑒𝑛2 (
𝑥
2
) ]
 
1
−2𝑠𝑒𝑛2(
𝑥
2)}
2𝑠𝑒𝑛2(
𝑥
2)
𝑠𝑒𝑛(𝑥)
 
𝑙 = 𝑒
lim
𝑥→0
−2 𝑠𝑒𝑛2(
𝑥
2 )
 2 𝑠𝑒𝑛(
𝑥
2)cos(
𝑥
2) = 𝑒
−lim
𝑥→0
𝑠𝑒𝑛(
𝑥
2 )
cos(
𝑥
2) = 𝑒0 = 1 
 
 
𝟑𝟏. 𝐥𝐢𝐦
𝒙→∞
𝒍𝒏(𝒄𝒐𝒔(𝒙))
𝒙𝟐
 
lim
𝑥→0
𝑙𝑛(cos(𝑥))
𝑥2
= lim
𝑥→0
𝑙𝑛(1 − 1 + cos(𝑥))
𝑥2
= lim
𝑥→0
𝑙𝑛[1 − (1 + cos(𝑥) ) ]
𝑥2
= 
lim
𝑥→0
𝑙𝑛 [1 − 2𝑠𝑒𝑛2 (
𝑥
2
) ]
𝑥2
= lim
𝑥→0
𝑙𝑛 [1 − 2𝑠𝑒𝑛2 (
𝑥
2
) ]
−2𝑠𝑒𝑛2(
𝑥
2)
−2𝑠𝑒𝑛2(
𝑥
2)
𝑥2
= 
 
 
 
 
 
lim
𝑥→0
−2𝑠𝑒𝑛2 (
𝑥
2
) 𝑙𝑛 [1 − 2𝑠𝑒𝑛2 (
𝑥
2
) ]
1
−2𝑠𝑒𝑛2(
𝑥
2)
𝑥2
= lim
𝑥→0
(
𝑠𝑒𝑛 (
𝑥
2
)
𝑥
2
)
2
∗ ln [1 − 2𝑠𝑒𝑛2 (
𝑥
2
) ]
1
−2𝑠𝑒𝑛2(
𝑥
2) = 
𝑙 = −
1
2
ln 𝑒 = −
1
2
 
 
 
𝟑𝟐. 𝐥𝐢𝐦
𝒙→𝟎
[
𝒔𝒆𝒏(𝒙)
𝒙
]
𝒔𝒆𝒏(𝒙)
𝒙−𝒔𝒆𝒏(𝒙)
 
lim
𝑥→0
[
𝑠𝑒𝑛(𝑥)
𝑥
]
𝑠𝑒𝑛(𝑥)
𝑥−𝑠𝑒𝑛(𝑥)
= lim
𝑥→0
[1 +
𝑠𝑒𝑛(𝑥)
𝑥
− 1]
𝑠𝑒𝑛(𝑥)
𝑥−𝑠𝑒𝑛(𝑥)
= lim
𝑥→0
[1 +
𝑠𝑒𝑛(𝑥) − 𝑥
𝑥
]
𝑠𝑒𝑛(𝑥)
𝑥−𝑠𝑒𝑛(𝑥)
 
lim
𝑥→0
{[1 +
𝑠𝑒𝑛(𝑥) − 𝑥
𝑥
]
𝑥
−𝑥+𝑠𝑒𝑛(𝑥)
}
𝑠𝑒𝑛(𝑥)
𝑥
= 𝑒
lim
𝑥→0
− 
𝑠𝑒𝑛(𝑥)
𝑥 = 𝑒−1 =
1
𝑒
 
 
 
𝟑𝟑. 𝐥𝐢𝐦
𝒙→𝟎
𝒍𝒏(𝒄𝒐𝒔(𝒂𝒙))
𝒍𝒏(𝒄𝒐𝒔(𝒃𝒙))
 
De acuerdo 
lim
𝑥→0
𝑙𝑛(𝑐𝑜𝑠(𝑥))
𝑥2
= −
1
2
 
entonces 
lim
𝑥→0
ln(cos(𝑎𝑥))
𝑙𝑛(cos(𝑏𝑥))
= lim
𝑥→0
𝑎2
𝑙𝑛(𝑐𝑜𝑠(𝑎𝑥))
(𝑎𝑥)2
𝑏2
𝑙𝑛(cos(𝑏𝑥))
(𝑏𝑥)2
= 
𝑎2
𝑏2
lim
𝑥→0
𝑙𝑛(𝑐𝑜𝑠(𝑎𝑥))
(𝑎𝑥)2
𝑙𝑛(cos(𝑏𝑥))
(𝑏𝑥)2
 =
𝑎2
𝑏2
(
−1
2
 
−1
2
=
𝑎2
𝑏2
= (
𝑎
 𝑏
 )
2
) 
 
𝟑𝟒. 𝐥𝐢𝐦
𝒙→−∞
𝒍𝒏(𝟏 + 𝒆𝒙)
𝒙
 
lim
𝑥→−∞
𝑙𝑛(1 + 𝑒𝑥)
𝑥
= lim
𝑥→−∞
𝑒𝑥
𝑥
ln(1 + 𝑒𝑥)
1
𝑒𝑥 = 𝑒 ∗ lim
𝑥→−∞
𝑒𝑥
𝑥
 
 
 
 
 
 
 
𝑙 = 𝑒 ∗ lim
𝑥→−∞
𝑒𝑥 = 𝑒 ∗ 𝑒−∞ = 𝑒 ∗
1
𝑒∞
= 𝑒 ∗ 0 = 0 
 
𝟑𝟓. 𝐥𝐢𝐦
𝒙→−∞
𝒍𝒏(𝟏 + 𝒆𝒙)
𝒙
 
lim
𝑥→+∞
𝑙𝑛(1 + 𝑒𝑥)
𝑥
= lim
𝑥→+∞
𝑙𝑛𝑒𝑥 (1 +
1
𝑒𝑥
)
𝑥
= lim
𝑥→+∞
𝑙𝑛𝑒𝑥 + 𝑙𝑛 (1 +
1
𝑒𝑥
)
𝑥
= 
lim
𝑥→+∞
[
𝑥 ∗ 𝑙𝑛𝑒
𝑥
+
𝑙𝑛 (1 +
1
𝑒𝑥
)
𝑥
] = 1 + lim
𝑥→+∞
𝑙𝑛 (1 +
1
𝑒𝑥
)
𝑒
𝑥𝑒𝑥
=1 +
𝑙𝑛 𝑒
lim
𝑥→+∞
𝑥. 𝑒𝑥
= 1 + 0 = 0 
 
𝟑𝟔. 𝐥𝐢𝐦
𝒙→𝟎
(√𝒙)
𝟑
𝟏+√𝟑𝒍𝒏(𝒙) 
𝑦 = (√𝑥)
3
1+√3𝑙𝑛(𝑥) → 𝑙𝑛(𝑦) = 𝑙𝑛(√𝑥)
3
1+√3𝑙𝑛(𝑥) 
𝑙𝑛(𝑦) =
3
1 + √3𝑙𝑛(𝑥)
𝑙𝑛 =
3
1 + √3𝑙𝑛(𝑥)
∗
𝑙𝑛(𝑥)
2
 
lim
𝑥→0
𝑙𝑛(𝑦) = lim
𝑥→0
3
1 + √3𝑙𝑛(𝑥)
∗
𝑙𝑛(𝑥)
2
 
lim
𝑥→0
𝑙𝑛(𝑦) = lim
𝑥→0
3
2
1
𝑙𝑛(𝑥)
+ √3
→ 𝑙𝑛(𝑦) =
3
2
0+ √3
=
3
2√3
=
√𝟑
2
→ 𝒚 = 𝒆
√𝟑
𝟐 
 
𝟑𝟕. 𝐥𝐢𝐦
𝒙→𝟎
𝒆𝜶𝒙 − 𝒆𝜷𝒙
𝐬𝐢𝐧𝜶𝒙 − 𝐬𝐢𝐧𝜷𝒙
 
lim
𝑥→0
(𝛼
𝑒𝛼𝑥 − 1
𝛼𝑥
) − (𝛽
𝑒𝛽𝑥 − 1
𝛽𝑥
)
(𝛼
sin 𝛼𝑥
𝛼𝑥
) − (𝛽
sin 𝛽𝑥
𝛽𝑥
)
 
lim
𝑥→0
(𝛼) − (𝛽)
(𝛼) − (𝛽)
= 1 
 
𝟑𝟖. 𝐥𝐢𝐦
𝒙→𝟎
𝐬𝐢𝐧𝟐𝒙
𝐥𝐧(𝟏 + 𝒙)
 
𝑒𝑥 = 1 + 𝑥; ln 𝑒𝑥 = ln(1 + 𝑥) 
lim
𝑥→0
sin 2𝑥
ln 𝑒𝑥
→ lim
𝑥→0
2 ∗ sin 2𝑥
2𝑥
= 2(1) = 2 
 
 
 
 
 
 
 
𝟑𝟗. 𝐥𝐢𝐦
𝒙→∞
(𝐬𝐢𝐧
𝟏
𝒙
+ 𝐜𝐨𝐬
𝟏
𝒙
)
𝒙
 
ℎ =
1
𝑥
; ℎ = 0; 𝑥 =
1
ℎ
 
lim
ℎ→0
(sin ℎ + cos ℎ)
1
ℎ 
lim
ℎ→0
(cos ℎ)
1
ℎ (1 +
sin ℎ
cos ℎ
)
1
ℎ
 
lim
ℎ→0
(1 + (−1 + cos ℎ))
1
ℎ(1 + tanℎ)
1
ℎ 
lim
ℎ→0
((((1 + (−1 + cos ℎ)))
1
cosℎ−1)
cosℎ−1
1
)
1
ℎ
∗ lim
ℎ→0
(((1 + tanℎ)
1
tanℎ)
tanℎ
)
1
ℎ
 
→ 𝑒
lim
ℎ→0
cosℎ−1
ℎ ∗ 𝑒
lim
ℎ→0
tanℎ
ℎ → 𝑒0 ∗ 𝑒 = 𝑒 
 
 
𝟒𝟎. 𝐥𝐢𝐦
𝒉→𝟎
𝒂𝒙+𝒉 + 𝒂𝒙−𝒉 − 𝟐𝒂𝒙
𝒉
, 𝒂 > 𝟎 
lim
ℎ→0
𝑎𝑥(𝑎ℎ + 𝑎−ℎ − 2)
ℎ
→ lim
ℎ→0
𝑎𝑥 ((
𝑎ℎ − 1
ℎ
) + (
𝑎−ℎ − 1
−(−ℎ)
))
ℎ/ℎ
 
𝑎−ℎ − 1 = 𝛼; 𝑎−ℎ = 𝛼 + 1; ln 𝑎−ℎ = ln(𝛼 + 1) ; −ℎ =
1
ln 𝑎
∗ ln(𝛼 + 1) 
lim
ℎ→0
𝑎𝑥 ((
𝑎ℎ − 1
ℎ
) − (
𝑎−ℎ − 1
−ℎ
))
ℎ/ℎ
→ 𝑎𝑥(2 ln 𝑎) → 𝑎𝑥 ln 𝑎2 
 
𝟒𝟏. 𝐥𝐢𝐦
𝒙→𝒂
𝒙 − 𝒂
𝐥𝐧𝒙 − 𝐥𝐧𝒂
 
lim.
𝑥→𝑎
𝑥 − 𝑎
ln
𝑥
𝑎
 
→
lim
𝑥→𝑎
 𝑥 − 𝑎
ln((lim
𝑥→𝑎
(1 +
𝑥 − 𝑎
𝑎
)
𝑎
𝑥−𝑎
)
𝑥−𝑎
𝑎
) 
→ lim
𝑥→𝑎
𝑥 − 𝑎
ln𝑒
𝑥−𝑎
𝑎
→ lim
𝑥→𝑎
𝑥 − 𝑎
𝑥 − 𝑎
𝑎
→ 𝑎 
 
𝟒𝟐. 𝐥𝐢𝐦
𝒙→𝟎
𝒆𝒂𝒙 − 𝟏
𝒃𝒃𝒙 − 𝟏
 
𝛼 = 𝑒𝑎𝑥 − 1; 𝑒𝑎𝑥 = 𝛼 + 1; ln 𝑒𝑎𝑥 = ln𝛼 + 1 ; 𝑥 =
ln 𝛼 + 1
𝑎
 
 
 
 
 
 
𝛽 = 𝑏𝑏𝑥 − 1; 𝑏𝑏𝑥 = 𝛽 + 1; ln 𝑏𝑏𝑥 = ln𝛽 + 1 ; 𝑏𝑥 =
1
ln 𝑏
(ln 𝛽 + 1) 
lim
𝑥→0
𝑒𝑎𝑥 − 1
𝑥
𝑏𝑏𝑥 − 1
𝑥
 
 
lim
𝛼→0
𝑎𝛼
ln 𝛼 + 1
lim
𝛽→0
𝛽
ln 𝛽 + 1
→
lim
𝛼→0
𝛼
ln(1 + 𝛼)
1
𝛼
lim
𝛽→0
𝑏𝛽
1
ln 𝑏
(ln 𝛽 + 1)
→
lim
𝛼→0
𝑎
lim
𝛽→0
𝑏
→
𝑎
𝑏
 
 
𝟒𝟑. 𝐥𝐢𝐦
𝒙→𝒃
𝒂𝒙 − 𝒂𝒃
𝒙 − 𝒃
. 𝒂 > 𝟎 
lim
𝑥→𝑏
𝑎𝑥 − 1
𝑥
𝑥 − 𝑏
𝑥
−
𝑎𝑏 − 1
𝑏
𝑥 − 𝑏
𝑏
→ lim
𝑥→𝑏
𝑥 ∗ ln 𝑎
𝑥 − 𝑏
− lim
𝑥→𝑏
𝑏 ∗ ln 𝑎
𝑥 − 𝑏
→
ln 𝑎 (𝑥 − 𝑏)
𝑥 − 𝑏
→ ln 𝑎 
 
𝟒𝟒. 𝐥𝐢𝐦
𝒙→𝟎
𝟏
𝒂𝒙
𝐥𝐧 √
𝟏 + 𝒂𝒙
𝟏 − 𝒂𝒙
𝟑
 
lim
𝑥→0
1
3𝑎𝑥
ln (
1 + 𝑎𝑥
1 − 𝑎𝑥
) 
lim
𝑥→0
1
3
ln
(
 
 
((1 +
2𝑎𝑥
1 − 𝑎𝑥
)
1−𝑎𝑥
2𝑎𝑥
)
2𝑎𝑥
1−𝑎𝑥
)
 
 
1
𝑎𝑥
 
1
3
ln
(
 
 
(lim
𝑥→0
(1 +
2𝑎𝑥
1 − 𝑎𝑥
)
1−𝑎𝑥
2𝑎𝑥
)
2𝑎𝑥
1−𝑎𝑥
)
 
 
1
𝑎𝑥
 
1
3
ln (𝑒
lim
𝑥→0
2𝑎𝑥
1−𝑎𝑥∗
1
𝑎𝑥) 
1
3
ln 𝑒
2
1−0 →
1
3
∗ 2 =
2
3
 
 
 
𝟒𝟓. 𝐥𝐢𝐦
𝒙→𝟎 
𝟓𝒙 − 𝟒𝒙
𝒙𝟐 − 𝒙
 
 
 
 
 
 
lim
𝑥→0 
(
5𝑥 − 1
𝑥
) − (
4𝑥 − 1
𝑥
)
𝑥(𝑥 − 1)
𝑥
 
ln 5 − ln 4
−1
= − ln
5
4
 
 
𝟒𝟔. 𝐥𝐢𝐦
𝒙→𝟏
𝒙𝒙 − 𝟏
𝒙 𝐥𝐧𝒙 
 
lim
𝑥→1
𝑥𝑥 − 1
𝑥
ln𝑥 
→ lim
𝑥→1
ln 𝑥
ln 𝑥
= 1 
 
𝟒𝟕. 𝐥𝐢𝐦
𝒙→𝟐
(
𝒙
𝟐
)
𝟏
𝒙−𝟐
 
. lim
𝑥→2
(1 +
𝑥
2
− 1)
1
𝑥−2
 
. lim
𝑥→2
(1 +
𝑥 − 2
2
)
1
𝑥−2
 
. lim
𝑥→2
(
 
 
((1 +
𝑥 − 2
2
)
2
𝑥−2
)
𝑥−2
2
)
 
 
1
𝑥−2
→ 𝑒
lim
𝑥→2
𝑥−2
2 ∗
1
𝑥−2 → 𝑒
1
2 → √𝑒 
 
𝟒𝟖. 𝐥𝐢𝐦
𝒙→𝟎
𝐥𝐧(𝐜𝐨𝐬𝒙)
𝐥𝐧(𝟏 + 𝒙𝟐)
 
lim
𝑥→0
ln (
sin 𝑥
tan 𝑥
)
ln(1 + 𝑥2)
→
ln(lim
𝑥→0
(((1 +
sin𝑥 − tan𝑥
tan 𝑥
)
tan𝑥
sin𝑥−tan𝑥
)
sin𝑥−tan𝑥
tan𝑥
))
ln (lim
𝑥→0
(1 + 𝑥2)
1
𝑥2)
𝑥2
 
ln 𝑒𝑙𝑖𝑚→
sin𝑥−tan𝑥
tan𝑥
ln 𝑒
lim
𝑥→0
𝑥2
→ lim
𝑥→0
cos 𝑥 − 1
𝑥2
∗
cos 𝑥 + 1
cos 𝑥 + 1
→ −lim
𝑥→0
1 − cos2 𝑥
𝑥2(cos2 𝑥 + 1)
 
−lim
𝑥→0
sin2 𝑥
𝑥2(cos2 𝑥 + 1)
→ −lim
𝑥→0
sin2 𝑥
𝑥2
∗ (
1
cos2 𝑥 + 1
) = −(1) (
1
2
) =
1
2
 
 
 
 
 
 
 
 
 
𝟒𝟗. 𝐥𝐢𝐦
𝒙→𝟎
𝐬𝐢𝐧𝟐 𝟑𝒙
𝐥𝐧𝟐(𝟏 + 𝟐𝒙)
 
ℎ = 3𝑥 𝑥 =
ℎ
3
 
lim
𝑥→0
sin2 3𝑥
ln2(lim
𝑥→0
(((1 + 2𝑥)
1
2𝑥)
2𝑥
)) 
 
sin2 3𝑥
ln2𝑒
lim
𝑥→0
2𝑥
 
→
sin2 ℎ
4
ℎ
9
2 →
9
4
lim
𝑥→0
sin2 ℎ
ℎ2
→
9
4
 
 𝟓𝟎. 𝐥𝐢𝐦
𝒙→𝟎
(
𝟏+𝒕𝒈 𝒙
𝟏+𝒔𝒆𝒏 𝒙
)
𝟏
𝒔𝒆𝒏 𝒙
 
= 𝑒 lim
𝑥→0
1
𝑠𝑒𝑛 𝑥
(
1+𝑡𝑔 𝑥
1+𝑠𝑒𝑛 𝑥
− 1)
,
 
= 𝑒 lim
𝑥→0
1
cos𝑥
(
−cos 𝑥3+1+sin𝑥3
cos𝑥+cos𝑥 sin 𝑥0
) 
𝑒0 = 1 
 
 𝟓𝟏. 𝐥𝐢𝐦
𝒙→∞
(
𝒂
𝟏
𝒙+𝒃
𝟏
𝒙
𝟐
)
𝒙
 
lim
𝑥→∞
(√𝑎
1
𝑥 + 𝑏
1
𝑥) 
=(√𝑎𝑏) 
 
𝟓𝟐. 𝐥𝐢𝐦
𝒙→∞
(𝐜𝐨𝐬
𝒂
𝒙
+ 𝒏 ∗ 𝐬𝐢𝐧
𝒂
𝒙
)
𝒙
 
= 𝑒 lim
𝑥→∞
(
1
cos
𝑎
𝑥
+𝑛∗sin
𝑎
𝑥
)
𝑥
 
 = 𝑒 lim
𝑥→∞
(
1
tan
sen
 (
𝑎
𝑥
) +𝑛∗sin
𝑎
𝑥
)
𝑥
 
= 𝑒 lim
𝑥→∞
(tan(𝑎) ∗ 𝑛)𝑥 
= 𝑒 lim
𝑥→∞
(a ∗ 𝑛) 
=𝑒𝑎∗𝑛 
 
 
 
 
 
 
 
 
 
𝟓𝟑. 𝐥𝐢𝐦
𝒙→∞
[𝟑 − 𝟐 (
𝒂𝒙+𝟏
𝒂𝒙
)]
𝐭𝐠(
𝝅
𝟐
(
𝒂𝒙+𝟏
𝒂𝒙
))
 
= 𝑒 lim
𝑥→∞
[1]
tg(
𝜋
2
(1))
 
 = 𝑒
4
𝜋 
𝟓𝟒. 𝐥𝐢𝐦
𝒙→𝟎
𝐥𝐧(𝒏𝒙 + √𝟏 − 𝒏𝟐𝒙𝟐)
𝐥𝐧(𝒙 + √𝟏 − 𝒙𝟐)
 
lim
𝑥→0
ln (𝑛𝑥 (
𝑛𝑥 + √1 − 𝑛2𝑥2
𝑛𝑥
∗
𝑛𝑥 − √1 − 𝑛2𝑥2
𝑛𝑥 − √1 − 𝑛2𝑥2
))
ln(𝑥 (
𝑥 + √1 − 𝑥2
𝑥
∗
𝑥 − √1 − 𝑥2
𝑥 − √1 − 𝑥2
))
 
lim
𝑥→0
ln (𝑛𝑥 (
2𝑛2𝑥2 − 1
𝑛𝑥(𝑛𝑥 − √1 − 𝑛2𝑥2)
))
ln(𝑥 (
2𝑥2 − 1
𝑥(𝑥 − √1 − 𝑥2)
))
 
lim
𝑥→0
= 𝑛 
 
 
 𝟓𝟓. 𝐥𝐢𝐦
𝒙→𝟎
(
(𝟏+𝐬𝐢𝐧 𝒙∗𝐜𝐨𝐬𝜶 𝒙)
𝟏+𝐬𝐢𝐧𝒙∗𝐜𝐨𝐬𝜷 𝒙
)
𝒄𝒕𝒈𝟑𝒙
 
= lim
𝑥→0
(
1+sin𝑥∗cos𝛼 𝑥
1+sin𝑥∗cos 𝛽 𝑥
)
𝑐𝑡𝑔3𝑥
 
= lim
𝑥→0
1(𝑒−𝛼2)
1+cos𝑥
 
= lim
𝑥→0
1(𝑒−𝛼2)
2
 
= lim
𝑥→0
𝑒𝛽2−𝛼2
2
 
=
𝑒𝛽2−𝛼2
2
 
 
𝟓𝟔. 𝐥𝐢𝐦
𝒙→𝒂
√𝒙
𝟑
− √𝒂
𝟑
√𝒂
+ (𝟐 −
𝒙
𝒂
)
𝒕𝒈(
𝒙𝒙
𝟐𝒂)
 
 
 
 
= lim
𝑥→𝑎
√𝑥
3
− √𝑎
3
√𝑎
+ (2 −
𝑥
𝑎
)
𝑡𝑔(
𝑥𝑥
2𝑎)
 
= lim
𝑥→𝑎
√𝑥
3
− √𝑎
3
√𝑥 − √𝑎
+ lim
𝑥→𝑎
(2 −
𝑥
𝑎
)
𝑡𝑔(
𝑥𝑥
2𝑎)
 
= lim
𝑥→𝑎
(√𝑥
3
− √𝑎
3
)(√𝑥2
3
− √𝑥𝑎
3
+ √𝑎2
3
)(√𝑥 + √𝑎)
(√𝑥 − √𝑎)(√𝑥 + √𝑎)(√𝑥2
3
− √𝑥𝑎
3
+ √𝑎2
3
)
 
= lim
𝑥→𝑎
(𝑥−𝑎)(√𝑥+√𝑎)
(𝑥−𝑎)( √𝑥2
3
− √𝑥𝑎
3 + √𝑎2
3
)
) + lim
𝑥→𝑎
𝑒
𝑎−𝑥
𝑎
𝑡𝑔(
𝑥𝑥
2𝑎
)
 
= lim
𝑥→𝑎
(√𝑥+√𝑎)
( √𝑥2
3
− √𝑥𝑎
3 + √𝑎2
3
)
+ lim
𝑥→𝑎
𝑒
𝑎−𝑥
𝑎
𝑡𝑔(
𝑥𝑥
2𝑎
)
 
= lim
𝑥→𝑎
2√𝑎
3√𝑎2
3 + lim𝑥→𝑎
𝑒
𝑡 𝐶𝑜𝑠 (
𝜋𝑡
2𝑎) 
𝑎∗𝑠𝑖𝑛 
=
2
3√𝑎2
6 + lim𝑥→𝑎
𝑒
 (
𝜋𝑡
2𝑎) 
sin(
𝜋𝑡
2𝑎) 𝑒
 2𝑐𝑜𝑠 (
𝜋𝑡
2𝑎) 
𝜋 
=
2
3√𝑎2
6 + 𝑒
1+
2 
𝜋 
 
𝟓𝟕. 𝐥𝐢𝐦
𝒙→𝟎
(
𝒂𝒙 + 𝒃𝒙 + 𝒄𝒙
𝟑
)
𝟏
𝒙
 
lim
𝑥→0
(1 +
𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥
3
− 1)
1
𝑥
 
lim
𝑥→0
(1 +
𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥 − 3
3
)
1
𝑥
 
lim
𝑥→0
 ((1 +
𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥 − 3
3
)
3
𝑎𝑥+𝑏𝑥+𝑐𝑥−3
)
𝑎𝑥+𝑏𝑥+𝑐𝑥−3
3𝑥
 
 
lim
𝑥→0
 𝑒
𝑎𝑥+𝑏𝑥+𝑐𝑥−3
3𝑥 
lim
𝑥→0
 𝑒
𝑎𝑥+𝑏𝑥+𝑐𝑥−1
3𝑥
 → lim
𝑥→0
 𝑒
𝑎𝑥−1
3𝑥
 ∗ lim
𝑥→0
 𝑒
𝑏𝑥−1
3𝑥
 ∗ lim
𝑥→0
 𝑒
𝑐𝑥−1
3𝑥
 
 Resolviendo cada limite: 
𝑟 = 𝑎𝑥 − 1 𝑎𝑥 = 𝑟 + 1 𝑥 =
ln(𝑟+1)
ln (𝑎)
 
 
 
 
 
𝑠 = 𝑏𝑥 − 1 𝑐𝑥 = 𝑠 + 1 𝑥 =
ln(𝑠+1)
ln (𝑏)
 
𝑡 = 𝑐𝑥 − 1 𝑐𝑥 = 𝑡 + 1 𝑥 =
ln(𝑡+1)
ln (𝑐)
 
 lim
𝑟→0
 𝑒
𝑟𝑙𝑛(𝑎)
3 ln(𝑟+1) ∗ lim
𝑠→0
 𝑒
𝑠𝑙𝑛(𝑏)
3 ln(𝑠+1) ∗ lim
𝑡→0
 𝑒
𝑡𝑙𝑛(𝑐)
3 ln(𝑡+1) 
lim
𝑟→0
 𝑒
𝑙𝑛(𝑎)
3 ln(𝑟+1) ∗ lim
𝑠→0
 𝑒
𝑙𝑛(𝑏)
3 ln(𝑠+1) ∗ lim
𝑡→0
 𝑒
𝑙𝑛(𝑐)
3 ln(𝑡+1) 
𝑒
𝑙𝑛(𝑎)
3 ln(𝑒) ∗ 𝑒
𝑙𝑛(𝑏)
3 ln 𝑒 ∗ 𝑒
𝑙𝑛(𝑐)
3 ln 𝑒 
𝑒
𝑙𝑛(𝑎)
3 ∗ 𝑒
𝑙𝑛(𝑏)
3 ∗ 𝑒
𝑙𝑛(𝑐)
3 
𝑒
ln(𝑎
1
3)
∗ 𝑒
ln(𝑏
1
3)
∗ 𝑒
ln(𝑐
1
3)
 
𝑎
1
3 ∗ 𝑏
1
3 ∗ 𝑐
1
3 
→ √ 𝑎𝑏𝑐
3
 
 
58. 𝐥𝐢𝐦
𝒙→𝟎
𝟖𝒙−𝟕𝒙
𝟔𝒙−𝟓𝒙
 
 
= lim
𝑥→0
8𝑥 − 1 + 1 − 7𝑥
6𝑥 − 1 + 1 − 5𝑥
 
 
= lim
𝑥→0
8𝑥 − 1 + 1 − 7𝑥
𝑥
6𝑥 − 1 + 1 − 5𝑥
𝑥
 
= lim
𝑥→0
8𝑥 − 1
𝑥
−
7𝑥 − 1
𝑥
6𝑥 − 1
𝑥
−
5𝑥 − 1
𝑥
 
=
ln8 − ln(7)
ln(6) − ln(5)
 
 
=
ln (
8
7
)
ln (
6
5
)
 
 
 
59. 𝐥𝐢𝐦
𝒙→𝟎
𝐥𝐧(𝟏+𝒙+𝒙𝟐)+𝐥𝐧(𝟏−𝒙+𝒙𝟐)
𝒙𝟐
 
 
 
 
= lim
𝑥→0
1
𝑥2
ln(1 + 𝑥 + 𝑥2) + lim
𝑥→0
1
𝑥2
ln(1 − 𝑥 + 𝑥2) 
= lim
𝑥→0
1
𝑥2
ln [(1 + 𝑥 − 𝑥2)
1
𝑥2+𝑥]
𝑥2+𝑥
+ 
= lim
𝑥→0
1
𝑥2
ln [(1 + 𝑥2 − 𝑥)
1
𝑥2−𝑥]
𝑥2−𝑥
 
 
= lim
𝑥→0
𝑥2 + 𝑥
𝑥2
ln [(1 + 𝑥 + 𝑥2)
1
𝑥2+𝑥] + lim
𝑥→0
𝑥2 − 𝑥
𝑥2
 𝑙𝑛 [(1 + 𝑥2 − 𝑥)
1
𝑥2+𝑥]
 
 
 
= ln(𝑒) lim
𝑥→0
𝑥 + 1
𝑥
+ ln(𝑒) lim
𝑥→0
𝑥 − 1
𝑥
 
= lim
𝑥→0
(
𝑥 + 1
𝑥
+
𝑥 − 1
𝑥
) 
= lim
𝑥→0
2𝑥
2
 
= 2 
 
 
 
60. 𝐥𝐢𝐦
𝒙→
𝝅
𝟐
(𝟏 + 𝒄𝒕𝒈𝒙)𝐬𝐞𝐜𝒙 
Cambio de variable 𝑡 = 𝑥 −
𝜋
2
 ; 𝑥 = 𝑡 +
𝜋
2
 
= lim
𝑥→
𝜋
2
 [1 +
cos (𝑡 +
𝜋
2
)
𝑠𝑒𝑛 (𝑡 +
𝜋
2
)
]
sec(𝑡+
𝜋
2)
 
= lim
𝑡→0
 [1 +
−𝑠𝑒𝑛 (𝑡)
𝑐𝑜𝑠(𝑡)
]
−csc(𝑡)
 
= lim
𝑡→0
 { [1 − 𝑡𝑔(𝑡)]
−
1
𝑡𝑔(𝑡)} −𝑡𝑔(𝑡) 
= 𝑒 lim
𝑡→0
1
cos (𝑡)
 
 =
1
𝑒