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					6x2		5x 1														10x2 9x 7
7.13.		lim											1.						7.14.	lim														19.
7.13.		lim				x 1 3							1.						7.14.	lim			x 7 5											19.
		x 1 3				x 1 3														x -7 5			x 7 5
						2x	2 13x 21										1											2x		2 9x 10 1
7.15.		lim																	.	7.16.		lim																				.
7.15.		lim						2x 7											.	7.16.		lim								2x 5												.
		x -7 2						2x 7									2				x 5 2									2x 5							2
					6x2		x 1																						6x	2 75x 39
7.17.		lim									5.									7.18.		lim																			81.
7.17.		lim				x 1 3					5.									7.18.		lim								x 1 2											81.
		x 1 3				x 1 3															x -1 2									x 1 2
				2x2			21x 11																	5x2						24x 5
7.19.		lim														23.				7.20. lim																	26.
7.19.		lim					x 11									23.				7.20. lim										x 5							26.
		x 11					x 11														x 5									x 5
				2x2			15x 7																		2x2					6x 8
7.21.		lim													13.					7.22.		lim														10.
7.21.		lim					x 7								13.					7.22.		lim								x 4						10.
		x -7					x 7														x -4									x 4
		lim			6x2 x 1												5	.		lim	x2 2x 15											8.
7.23.																			7.24.
7.23.						3x 1													7.24.				x 5
		x -1 3				3x 1											3			x -5			x 5
			3x2 40x 128																							5x2				51x 10
7.25. lim																	8.			7.26.		lim																		49.
7.25. lim							x 8										8.			7.26.		lim								x 10										49.
		x 8					x 8														x 10									x 10
7.27.		lim			2x2		5x 2							3.						7.28.		lim			3x2					17x 6								19.
7.27.		lim				x 1 2								3.						7.28.		lim								x 6								19.
		x 1 2				x 1 2															x -6									x 6
					3x2		17x 6																				15x2 2x 1
7.29.		lim														19.				7.30.		lim																	8.
		x 1 3				x 1 3															x -1 5									x 1 5
				15x			2 2x 1
7.31.		lim													8.
7.31.		lim				x 1 3									8.
		x 1 3				x 1 3
		Задача 8. Доказать, что функция																	f x непрерывна в точке				x0 (найти ).
8.1.		f x 5x2						1,		x0				6.						8.2.	f x 4x2										2,				x0				5.
8.3.		f x 3x2						3,		x0					4.					8.4.	f x 2x2										4,				x0				3.
8.5	f x 2x2								5,			x0				2.				8.6 f x 3x2												6,				x0 1.
8.7	f x 4x2								7,			x0				1.			8.8 f x 5x2 8,													x0 2.
8.9	f x 5x2								9,			x0				3.			8.10	f x 4x2 9,												x0 4.
8.11		f x 3x2 8,											x0				5.			8.12	f x 2x2												7,				x0 6.
8.13		f x 2x2							6,		x0					7.			8.14	f x	3x2 5,										x0		8.


8.15	f x 4x2	4,		x0	9.		8.16	f x 5x2		3,	x0	8.
8.17	f x 5x2	1, x0			7.		8.18	f x 4x2		1,	x0	6.
8.19	f x 3x2	2,		x0	5.		8.20	f x 2x2		3,	x0	4.
8.21	f x 2x2		4,	x0		3.		8.22	f x	3x2		5,	x0	2.
8.23	f x 4x2		6,	x0		1.		8.24	f x	5x2		7,	x0	1.
8.25	f x 4x2		8,	x0		2.		8.26	f x	3x2		9,	x0	3.
8.27	f x 2x2		9,	x0		4.		8.28	f x	2x2 8,			x0 5.
8.29	f x 3x2	7,		x0	6.		8.30	f x 4x2		6,	x0 7.
8.31	f x 5x2	5, x0			8.

Задача 9. Вычислить пределы функций.

x3 2x 1 x 1

9.1. lim			.
	x4 4x2
x 1		5

x2 3x 2 2

9.3. lim		.

x 1 x3 2x2	x 2

x2 2x 3 2

9.5. lim		.

x 3 x3 4x2	3x

1 x 3 (1 3x)

9.7.

lim

x x5

x 0

9.9.

lim

3x 2

x2 x 2

x 1

x3 3x 2

9.11. lim

x 1 x3

x2 x 1

x3 4x2 5x 2

9.13.

lim

x 1

x3 3x 2

5x2 8x 4

9.15. lim

x3 3x2

x -2

6x2 12x 8

9.17. lim

x3 3x2

x 2

9.2. lim	x3	3x 2	.
		x x2
x 1

2x2 x 1 2

9.4. lim		.

x 1 x3 2x2	x 2

	x3	2x 1 2
9.6. lim			.

x 1 x4		2x 1

9.8.lim x2 2x 1.

x1 2x2 x 1

5x2 7x 3

9.10.

lim

x 1 x3

4x2 5x 2

5x 3

9.12. lim

x 1 x3

x 1

x4 1

9.14. lim

x 1 2x4 x2 1

5x2 8x 4

9.16. lim

x 2

x3 3x2 4

x3 5x2 8x 4

9.18.

lim

x -2 x3

7x2 16x 12

9.19.

lim

x3 3x 2

2 x 2)2

x 1 (x

3x 2

9.21.

lim

x 1 x2

2x 1

x4 1

9.23. lim

x 1 2x

x2 1

9.25. lim

2x2 x 1

x 1 x3 2x2 x 2

2x 1

9.27.

lim

x 1 x4

2x 1

2 1

9.29. lim

x 1 2x

x 1

4x2 3x 18

9.31. lim

x 3 x3

5x2 3x 9

Задача 10. Вычислить пределы функций.


10.1	lim		1 2x 3						.
10.1	lim								.
	x 4					x 2
							.
10.3	lim			x 1
10.3	lim
	x 1 3		x2 1
		3						2
		3			x 6			2
10.5	lim									.
10.5	lim									.
	x 2				x3 8

10.7 lim	9		2x 5	.

x 8	3	x 2

	3	8 3x x2								2
10.9	lim												.
10.9	lim		x x2										.
x 0			x x2
		3				1
10.11	lim	3			x	1					.
10.11	lim										.
	x 1	1 x						2x
			3				2
10.13	lim		3	4x			2					.
10.13	lim											.
	x 2		2 x						2x

x3 3x 2

9.20. lim

x 2

x2 2x 1

9.22. lim

x 1 x3

x2 x 1

9.24. lim

x2 3x 2

x 1 x3 2x2 x 2

2 2x 3

9.26. lim

x -3 x3

4x2 3x

(1 x)3 (1 3x)

9.28. lim

x2 x5

x 0

7x2 15x 9

9.30. lim

x -3 x3

8x2 21x 18

10.2.		lim				1 x							3	.
10.2.		lim												.
		x 8				2 3					x
					2											.
10.4 lim		x 13			2				x 1
10.4 lim			x2			9
x 3			x2			9
			4				2
10.6		lim	4		x		2			.
10.6		lim								.
	x 16				x 4

				1 2x x2 (1 x)
10.8		lim																			.
10.8		lim											x								.
	x 0												x
			3									3
			3		27 x							3						27 x
10.10		lim																		.
10.10		lim																		.
		x 0					x 23										x4
																			.
10.12		lim				1 x									1 x
10.12		lim
		x 0 3 1 x 3 1 x
							1	.
10.14		lim			x		1
10.14		lim
		x 1		x2 1

			3 9x 3
10.15	lim										.
10.15	lim										.
	x 3	3 x							2x
		3					4
		3		16x			4
10.17	lim											.
10.17	lim											.
	x 4	4 x								2x
		3						1 2
		3				x 4		1 2
10.19	lim												.
10.19	lim												.
	x 1 2	1 2 x 2x
			3					1 4
			3		x 16			1 4
10.21	lim												.
10.21	lim												.
	x 1 4	1 4 x 2x

10.23 lim 327 x 327 x .

x 0 3 x2 5x

			1 2x 3x2 (1 x)
10.25	lim															.
10.25	lim															.
	x 0						3					x
												x
		4							2
10.27	lim	4						x	2				.


	x 16	3 (						x	4)2
						2					.
10.29	lim				x	2
10.29	lim
	x 4	3 x2 16
										2					.
10.31	lim			x 13						2				x 1
10.31	lim
	x 3							3 x2 9

Задача 11. Вычислить пределы функций.

11.1. lim	ln 1 sin x	.

x 0	sin4x

11.3lim 3x2 5x.

x0 sin3x

	4x
11.5 lim		.

x0 tg( (2 x))

1cos3 x

11.7lim 2 .

x 0 4x

10.16 lim 3x 6 2.

	x -2		x 2
						5		.
10.18	lim	9 2x

	x 8	3 x2 4
				3			1 3
					x 9
10.20	lim								.

	x 1 3	1 3 x 2x

10.22	lim	1 x						1 x			.
10.22	lim										.
	x 0			7		x
						x
	3
	3		8 3x x2							2
10.24	lim											.
10.24	lim											.
	x 0			3 x2 x3
							5		.
10.26	lim		9 2x				5
10.26	lim	3
	x 8	3			x 2
					x 2

10.28 lim 3x 6 2.

x -2 3 x3 8

10.30 lim 10 x 61 x .

x 8	2	3	x

11.2. lim 1 cos10x.
x 0	ex2	1

11.4	lim		1 cos2x			.

	x 0 cos7x cos3x
		2x
11.6 lim				.

x 0 tg[2 (x 1 2)]
11.8	lim		arcsin3x		.

	x 0		2 x 2

		2x 1
11.9	lim		.

x 0 ln(1 2x)
11.11	lim	ln(1 7x)		.

x0 sin( (x 7))

11.13lim 9ln(1 2x).

x0 4arctg3x

11.15	lim	sin7x		.

	x 0	x2 x
11.17	lim	2sin[ (x 1)]					.

	x 0	ln(1 2x)
					1	.
11.19	lim		1 x

x0 sin[ (x 2)]

11.21lim 1 cosx .

x 0 xsin x

	e4x 1
11.23 lim		.

x0 sin( (x2 1))

11.25lim sin2 x tg2x.

x0 4x

11.27 lim tgx sin x .

x0 x(1 cos2x)

11.29lim tg( (1 x2)).

x 0

ln(x 1)

11.31lim 2xsin x .

x01 cosx

Задача 12. Вычислить пределы функций.

12.1.lim x2 1.

x1 ln x

12.3 lim 1 cos3x. x sin2 7x

11.10 lim	arctg2x	.

x0 sin(2 (x 10))

11.12lim cos(x 5 2)tgx.

x0 arcsin 22x

11.14	lim			1	3x 1				.

	x 0 cos[ (x 1) 2]
					2	.
11.16	lim			4 x

	x 0		3arctgx
11.18	lim		cos2x cosx				.

	x 0			1 cosx
11.20	lim	sin[5(x )]						.

	x 0			e3x 1

11.22lim arcsin2xln2.

x0 2 3x 1

11.24	lim	1 cosx		.

	x 0 (e3x 1)2
11.26	lim	arcsin2x				.

	x 0 ln(e x) 1
		ln(x2 1)
11.28	lim				.

	x 01 x2		1
11.30	lim	2(e x	1)				.

x0 3(3 1 x 1)

12.2.lim x2 x 1 1.

x 1	ln x
12.4 lim	1 sin2x	.

x 4 ( 4x)2

12.5 lim 1 cos x.
x 1	tg2 x

		sin2 x tg	2x
12.7	lim			.
		(x )4
	x
12.9	lim	cos5x cos3x			.

	x	sin2 x

12.11lim sin7 x.

x2 sin8 x


				x2 3x 3 1
12.13	lim								.

	x 1				sin x
			5x 3			2x2
12.15	lim	3			3		.

	x 1			tg x
12.17	lim			ln2x ln						.

	x 2 sin(5x 2)cosx
12.19	lim			e		ex		.

xsin5x sin3x

124 x2

12.21lim .

x2 2(2x 3x2 5x 2)

12.23lim tg x.

x-2 x 2

12.25 lim 1 2cosx.

x 3

12.27lim 1 x2 .

x1 sin x

12.29 lim 3 10 x . x 1 sin3 x

12.31 lim cos3x cosx.
x	tg2 2x

Задача 13. Вычислить пределы функций.

12.6lim tg3x.

x2 tgx

12.8lim x2 x 1 1.

x 1	tg x
			sin7x sin3x
12.10	lim										.
12.10	lim			ex	2	e4			2		.
	x 2			ex		e4
12.12	lim			ln(5 2				x)		.
12.12	lim									.
	x 2			10 3x 2
12.14	lim	x2				2	.
12.14	lim		sin x				.
	x		sin x

12.16lim 2x 16.

x4 sin x

12.18 lim lntgx .

x4 cos2x

12.20lim ln(9 2x2 ).

x 2

sin2 x

12.22lim x 1.

x1 4 x 13

12.24	lim		1 sin(x 2)	.

	x		x
		arctg(x2 2x)
12.26	lim				.

	x 2		sin3 x

12.28 lim cos( x2).

x 1

1 x

12.30lim sin5x.

xtg3x

13.1.	lim						2cos2 x 1					.
13.1.	lim							lnsin x				.
	x 2							lnsin x
									ln x 3
	lim								ln x 3								2x 3								.
13.3
13.3
x 2 sin x 2													sin x 1

							etg2x e sin2x
13.5.	lim																	.
13.5.	lim																	.
	x 2								sin x 1
		sin
										2x2		3x 5
										2x2		3x 5												1 x
13.7. lim																										.
13.7. lim						ln x 1							ln x 1													.
	x 3					ln x 1							ln x 1										ln2
13.9.	lim								ln(4x 1)												.
13.9.	lim																				.
	x 1 2							1 cos x 1
									2sin x 1
13.11	lim																		.
13.11	lim																		.
	x 3 ln(x3 6x 8)
13.13	lim			tgln(3x								5)		.
13.13	lim			2										.
	x 2					ex 3 ex					1

				3 1 ln2 x 1
13.15	lim														.
13.15	lim														.
	x 1						1 cos x
13.17	lim		ln(2x 5)										.
13.17	lim												.
	x 3					esin x 1
							esin2x etg2x
13.19	lim															.
13.19	lim															.
	x 2								ln(2x )

								2x 7 2x 1 5
13.21	lim																					.
13.21	lim																					.
	x 1									x3 1
					(x3 3)sin5x
13.23	lim																			.
13.23	lim																			.
	x								esin2 x 1

13.25lim lncos2x.

xlncos4x

13.27lim ax2 a2 1 .

xa tgln(xa)

13.2.	lim				2x 1						2						.
13.2.	lim																.
	x 1 2 esin x e sin3 x
13.4.	lim		tgx tg2									.
13.4.	lim											.
	x 2 sinln x 1
13.6.	lim			lnsin3x						.
13.6.	lim									.
	x 6 6x 2
13.8.	lim			x 2 2												.


	x 2 tg cosx 1
13.10	lim		arcsin(x					2) 2												.


	x -2				3 2 x x2					9
13.12	lim		lncos2x						.
13.12	lim								.
	x (1 x)2
13.14 lim	lncosx					.
13.14 lim						.
x 2	3sin2x 1
13.16	lim			cos(x 2)									.
13.16	lim												.
	x esin x esin4x
13.18	lim			esin2 6x esin2 3x															.
13.18	lim																		.
	x 3 log3 cos6x
			tg(ex 2 ex2 4 )
13.20	lim																	.
13.20	lim				tgx tg2													.
	x -2				tgx tg2
13.22	lim	ln(2 cosx)												.
13.22	lim													.
	x			(3sin x 1)2
13.24	lim				tg(x		1)								.


	x -1 e3 x3 4x2 6 e
13.26	lim				lnsin x						.
13.26	lim		2 (2x )2								.
	x		2 (2x )2

13.28 lim sin(e31 x22 e3x 2 ).

x -3	arctg(x 3)

13.31 lim sin(x2 ) .

x 2 sin x 1 2

Задача 14. Вычислить пределы функций.

14.1. lim		72x 53x				.
14.1. lim						.
x 0	2x arctg3x
		62x 7 2x
14.3. lim				.
14.3. lim				.
x 0 sin3x 2x
14.5. lim		32x 53x			.
14.5. lim					.
x 0 arctgx x3
14.7. lim		35x 2x	.
14.7. lim			.
x 0	x sin9x
14.9. lim		12x 5 3x					.
14.9. lim							.
x 0 2arcsin x x
14.11. lim		35x 27x						.
14.11. lim								.