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Пензенский Государственный Университет Архитектуры и Строительства

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III

4. ЗАДАНИЯ К ТИПОВЫМ РАСЧЕТАМ

4.1. Введение в анализ. Пределы

Контрольная работа ¹ 1. Вычисление пределов
			Вариант 1
lim		3n2 2n3 7	lim		2n3 n4 5
		7n 4n3 3
n			n 9n3 n2 8
lim		(2n 3)3 n3	lim	5n4 3n2 n3
		10n2 2n3			4n3 n4 10
n			n
lim		3x2 6x 45	lim			x3 5x2
		x3 25x			3x2 6x 45
x 5			x 5
lim	3x2 12x 12		lim			(x2 6x)2
		(x3 8) 3				x3 36x
x 2			x 6

lim

2x 2

x 1

lim

2 x x

x3 2x2 3x

x3 8

x 3

x 2

lim

3x 1 2

lim

x2 2 x2

x2 6x 5

x3 2x

x 1

x 2

lim

x arcsin 3x

lim

x2 cos3x

x 0 12x2 tg2x

x 0 1 cos4x

lim

2x3 x sin 3x

lim

2 tg3x x2

x2 sin 2x tg2 3x

x arctg2x

x 0

n 5

5 3n n 3

lim

n n 3

2 3n

3 4x

3x 2

lim

2x 5

x 3

lim

3x 3

x 3

x 0

3n2 1

n 2

lim

5 n

lim n ln

lim x2 ctg2 5x

lim

3 x

2 3x

x 0

III

Вариант 2

lim		n2 6n5 13
			2n4 n3 5
n
lim		5n4 6n7 1

n 12n5 7n6 n7
lim			2x2 8x 6
			6x3 54x
x 3
lim	3x2 6x 24
			4x (x2 4)
x 2

lim

1 x 1

3x2 4x

x 0

lim

x2 6x 8

x 4

lim 3x arcsin 2x

x 0

4x2 8x

lim

(2x 3) sin 4x

(x2 9) x cos3 x

x 0

lim

n 2

n 4

4 x

lim

x 0

5 6x

3n 2n2

2n 3

lim

n 1

lim 1 sin 2x

2 3x

x 0

lim n3 4 2n4 n2 n 3n n 10

lim 12n9 4n8 3 n 3n5 10n10


lim		6x3 12x2
	2x2 18x 28
x 2
lim	x2 10x 25
			5) (x 5)
x 5 (x
lim	6 2x			3 x
		4x (x2 1)
x 1
lim		sin 3x
		1	2x 1
x 0

lim 1 cos 6x x 0 x arctg5x

lim	13x2	x arcsin 4x
lim		tg2 5x
x 0		tg2 5x

lim 2n 5 n 1 n 2n 10

2 x

lim 4x 13 x 3

x 3

lim n	2	n2		1
		ln
		ln	n	2
n			n
n

lim 2 3x ctg2x x 0 5x

Вариант 3


lim				4n4	625
				3n3	5n4
n
lim	2n6 5n7 18
		2		3n5 4n8
n
lim			3x2 9x 6
				2x3 8x
x 2

lim x3 1

x 1 x3 x

lim				4 x x 2
				x3 9x
x 3
lim		4 x			2x 1
				5x x2
x 5
lim				sin 2x 3
				6x2 9x
x 3
		2
lim	x2		3x arcsin 2x
				2x3 5x2
x 0

lim x 2 2 x 3 x x 3

		3x 1	5 2 x
lim			x

x 0		1 2x
	n3	n2	3
lim		lg

n	3	n2	5

3x2 4

lim 5 2x 3x

x 2

lim

0,001n5 2n4

3n4 4n3 107

lim

(2n 3)2

4n2

18n 1000

lim

2x 3 2 4x2

16x2 8x 15

x 3

lim

3x2 18x 24

x3 8

x 2

lim

4 3x 2 x

tg5x

x 0

lim

( 2x 1)2

3 2x 4x

x 1

lim

x3 1

x 1 tg x 1

lim

sin 3x 2

10x 15x2

x 2

lim 3x 3

4 3 x

x 3

2 x

lim

x 2

n2 1

lim

n 2

lim 2x 3 x 6 x 3x 2

III

Вариант 4

lim

107 n3

n 2n3 3n 1

lim

5n4 3n2

7n3 10n2

lim

5x2 10x 15

2x3 18x

x 3

lim

5x2 20x 20

6x cos x 2

x 2 3x2

lim

4 x

x2 16

x3 25x

x 5

lim

16x2 8x 3

4x 7 4x 1

x 3

lim

2x3 x tg5x

4x2 3x3 x

x 0

lim

4x3 sin 2x 6

x3 3x2

x 0

lim

2n 3

2x 4

x2 5

lim

4 5x

x 0

2 3x2

3x2

lim

2x 1

x |5

lim 1 2tg3x x

x 0

lim

4n2 7n5 10

6n5 3n2 1

lim

2n 3 3 2n3

21 3n2 6n3

lim

2x3 3x2

4x2 12x 9

x 3

lim

100x2 10x 2

10x3 2x2

x 0,2

lim

4 3x 2 x

tg5x

x 0

lim

4x2 8x 4

3 x 2x

x 1

lim arcsin 3x

x 0 5x2 4x

lim

12x2 10x 2

(2 x) tg(1 2x)

x 2

l im

4x 11

x 3

2 x 3

lim

3x2 11 x 2

x 2

lim

1 x

lim n

n 3

III

Вариант 5

lim

5n2 6n4 107

3n4 2n 13

lim

2n5 3n10 10 9

3n7 10n6 12

lim

25x2 5x 2

25x3 4x

x 2

lim

2x3 16

2x3 27x

x 3

lim

5 2x 1

x 2 3x2 12x 12

lim

5x 6

15x

15x2 25x3

x 3

lim

x tg5x 3x2

arcsin4x

x 0

lim 2x arctg(x 2)

x 2

3x2 6x

lim

3n 5

2n 1

2 x 3

lim

5 4x

5 x

x 0

5x2

10x

2 1

lim

x x 1

2n 5

3n 1

lim

n 10

lim

2x8 x6 105

109 x7 6x

lim

4x4 6x5 106

4x5 x6

lim

10x3 10x

2 12x 15

x 1 x

lim

x3 144x

2x2 10x 24

x 12

lim

3x2 6x 3

5x 4 3x

x 1

lim

3 arctg2x

4 3x 4 x

x 0

lim

5x3 arctg2x

arcsin5x

x 0

lim

5tg2 (3x 4)

2 24x 16x

x 4

x 3

2 x

x2 8

x 3

lim

2 x 1

lim

1 sin 5x x

x 0

lim n

3n 1

4 x2

lim 5x 7 2 x 3

x 1,5

III

Вариант 6

lim		3n2 6n5 107
lim			10n5 2n4 9
n			10n5 2n4 9
lim		3n2 1 2 6n3
n				2n2 10n4 5
lim					12x2 8x 7
lim			(12x 3)2 16
x 1			(12x 3)2 16
	2
lim	2x 3 2 4x 6
lim		12x2 4x 65
x 5		12x2 4x 65
2
lim						5 2x 4x
lim							6
x 0,5 4x2 10x							6
lim						3x 2	6x
lim						9x2 4
x 2						9x2 4
	3
lim						3x2 tg5x
lim					5x arcsin2x
x 0					5x arcsin2x

lim	3x 6 2 cos x 2
x 2	x		2	2	x
	sin	3

lim n 2 n 1 n n 3

3 x 1

lim 2x 7 x 3

x 3

	2x3 3	6x2 10
lim	x2	3x
x	x2	3x

lim	3		2 3x
lim	x	lg
x 0	x		2 x

lim 3n6 4n5 13 n n3 2n4 4n5

lim

n 2 3 n3

2n 3n3 7

lim

(3x 5) 2

(3x2 12) x

x 2

4x sin

lim

(5x 1) 10x 1

x 0

lim

2x 3 2 4x2

16x2 8 1

x 3

3 arcsin8x

lim

1 5x3 3x 1

x 0

lim

3x arctg x 2

x 1

2x 2 sin

lim

2x arctg8x

x2 4 sin5x

x 0

	2x 3	x 6
lim
lim	2x 5
x	2x 5
	3 2x	4 x
lim	3 2x	x
lim	3 5x
x 0	3 5x
		3 x 4
lim x 2		x
x 4
		2x 5

lim 4 3x x 2

x 1

III

Вариант 7

lim

(n 3)3 n3

2n3 3n2 1

lim

(2n 3)3 5n3

4n2 10n3 3

lim

10x2 15x 10

4x2 8x

x 2

lim

3x2 6x 8

9x2 16

x 4

lim

3x 8 3

18x2 2

x 1

lim

1 3x 1

5x3 10x

x 0

lim 3x arcsin 2x

x 0

4x2 3x

lim

x arctg(2x 1)

24x2 6

x 0,5

lim

2n 1

2n 3

3 x


	lim		x 6			x 5
	x 5
	3x2 x				12x2		1
lim	x 2					4x 1
x	x 2					4x 1
lim n		2		n 2
lim n			lg			n
n						n

lim 4n5 3n4 9 n 107 5n3 n5


lim			3n2 1 4n3
n 2n 1 3 4n3
lim	3x2					9x x 32
							x3 9x
x 3
		lim					x sin 5x

		x 0					2x2	3x
lim					5x2 6x 8
							6 x x
x 2
							2x
							tg
lim								3
			2			3x		2 x
x 0
lim				x tg3x 4x

x 0								3x
							sin
								2

lim x 3 tg x 3
x 3	2x2 12x 18
	x 5			3 x2
	x 5				x
lim
lim			x
x			x
		2 3x			4 2 x
lim		2 3x			x

		2
x 0		2	5x
					x2 3
lim 3x 4					x
x 2
					2 3 x
lim	1 sin x 4 x
x 0

III

Вариант 8

lim 4n23 3n3 100 n 5n 10n 13

lim

2n3 3n4 10

lim

3x2 4x 10

2x x2

x 1

lim 4

12x 10 2x2

2 x2

x 1

lim

5 2x x 1

3x2 6x

x 2

lim

2x3 3x

2x 4 x

x 0

lim

tg5x 8x2

x 0 arcsin2x 3x

lim

2x sin 3x

x 0

4x2 5x3

lim

n 2

2 x

lim

3 x

x 0

3 5x

3n2

6n3 2n

lim

2n 3

6n2 1

x 3

lim

1 tg2x

x 0

lim

4 (n 5)3

3n4 n3

lim

n2 (n 2)2

4n 100

lim

16 6x x2

6x 3x2

x 2

lim

6x3 6x2

3x2 12x 9

x 1

lim

x3 2x2 3x

3x 1 2x

x 1

lim

4x2 16x 15

2x 6 2x

x 3

lim

3x2 x tg2x

2x3 -arcsin5x

x 0

lim

3x x sin 2x

x 0

2x2 -arctgx

lim

2 x

lim 2x 7

x 3

2x 1

4 x

x 2

lim

2x 5

lim n

n 1

III

Вариант 9

lim

5n 4n2

2n2 3n3 9

lim

(n 4)2 2n2

3n 5n2

lim

12 3x 3x2

5 10x2 5x

x 2

lim

3x2 3x3

x 1 x2 6x

lim

sin 3x

4 2x x 4

x 0

lim

3x 10

6x 8

9x2 18x 4

x 2

lim

3x2 arctg2x

4x 5x2

x 0

lim

8x2 x arcsin 5x

1 cos 2x

x 0

lim n 1 n 4 n n 2

	3x 2		4 x
lim	3x 2		x
lim
x 0	2 x
	2x 3	1 x2
lim	2x 3		x
lim	x 1
x	x 1

lim x 2 ctg 2 x

x 2

lim	4 (n 8)3

n 3n2 4n3 2

lim

4n2 5n 1

n n 3 2 n2

lim

2x2 12x 10

2x2

10x

x 5

lim

4x2 12x

3x2 12x 9

x 3

lim

5 4x x

x 1 2x2 6x 4

lim

2x3 18x

x 6 x

x 3

lim

x arcsin 4x

x 0

tg2 5x

lim

x arcsin 4x

x 0

tg2 5x

lim

3 x

lim

3x 5

x 2

2n2

6n3

lim

3n2

n n

lim n

n 1

III

Вариант 10

lim

2n4

15n3 9

3n5 4

lim

3n5

6n4

4n3 5n4 6

lim

4x 12

3x3 6x2

x 2

lim

6x 27

x 3

lim

x 6

3 2x

9x x3

x 3

lim

6 x x

3x2 6x

x 2

lim

x tg2x 3x2

4x arcsin3x

x 0

lim

3x3 3 arctg2x

x cos 2x 3x2

x 0

lim

3 n

n 2

2x 5

2 x 3

lim

x 0

	2n2 5n	4n3
lim
lim	3n 4	6n2 5
n	3n 4	6n2 5

lim x ctg3x

x 0

lim			2n2 3n4 1
				5n4 6n2
n
lim		(n 4)3 3n3
			4n2 6n3 2
n
lim				5x3 5x

x 1 3x2 6x 3
lim	x2 6x 8 2
				x 3x 12
x 4
lim				x2 4x
				x	8 x
x 4
lim				5 4x 2x
				x3 1
x 1
lim6				2x sin2 3x
				2x3 3x2
x 0
lim				x3 -tg2 3x
	arcsin2x 3x2
x 0 x

lim 2n 5 n 2 n 2n 7

2 x 1

lim 2x 9 x 4

x 4

		2n	6n2
lim
			3n 2
n
lim	3		2x 3
	x	ln	3 5x
x 0

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