Tipovye_raschyoty_po_vyssh_matematike_Ch_1_Uch

3

2

x

2x 2

2x 7

5 4x

7 x

3.22. lim

x

(x2 x 2)tg

.

4

2

x

x

3

x 1

1 x

4

x 1

3

2

x

3x

3x 2

x 3 1

x

(x 2)tg

3.23.

lim

2

.

2x

x 6

2 x

x 6

4

x 2

3

2

2

3x

2x

x

x 9 10 x

lim

x

3.24.

4 x 3x2

1 x

x 1

7 x

(x2 x 2)tg

.

2

3

2

x

5x

8x 4

2x 5

x 7

(x2 4)tg

9 x

3.25. lim

.

3

2

2

x

3x

4

x

2x

4

x 2

3

2

x

3x

4

3x 7

3 x

(x2

3 x

3.26.

lim

x 2)tg

.

3

5x

2

8x 4

x2 2x

4

x 2

x 2 x

2

5x

8x 3

17 8x

2x 7

9 x

3.27. lim

(x 1)tg

2

2

.

2x

3x 1

x

x

2

x 1

3

x

2

5x 5

x

2

x 2

9 x

x

(x 1)tg

3.28.

lim

.

x

2

3x 2

5 x2

2

3x 7

x 1

3

2

2x

x

4x 3

2x 1

7x 6

5 (x2

5 x

3.29. lim

1)tg

.

3

2

x

2x 1

x

2x 3

2

x 1

3x

3

13x

2

16x 4

x

2

2x 8

7 (x2

7 x

3.30. lim

x 6)tg

.

3

2

x

x

x 2

x 2

6 x

4

x 2

Задание 4.

При

x 0

определите

порядок

малости

бесконечно

малых

функций

(x), (x)

и (x)

относительно функции x. Обоснуйте, справедливо ли равен-

ство (x) ( (x))

при x 0.

tg(x2

3

3

x)

4.1. (x) 6sin

x,

(x) 2

,

(x) 5

arcsin(x 2 )

1.

3x 1

x

4.2. (x) 5arctg3

,

(x) 2e3x 4 x2,

(x) 3sin

cos2x.

4x2

2

51

	4.3. (x) esin3 x 1,	(x) 5e x		arctg3x,		(x) 3x2		2x 1 3.
	4.4. (x) 1 etg(x3),		arcsin3	x	, (x) 4ln1			.
		(x)					x7
			2x 3

4.5.(x) log2(1 tg2 x3)15 ,

4.6.(x) 2x sin(17x10),

(x) 31 arcsin3(x2) 1.

4.7. (x) 9 3x2 2,

(x)

(x) 71 x ln(1 3x2), (x) tg9x3 .

	(x) e5x					ln(1 9x7),
		arctg		x5		, (x) e9x ex.
		4	7x

4.8. (x) 1 5x2

4x3

1,

(x) (x2

5)arctg(x4), .

(x) ex2 sin3 (3x) 1.

4.9. (x) 5arctg3(

(x) 3sin(x2) cos(x4),

(x) xtg2

x

4x),

.

2

4.10. (x) earcsin2 (

) 1,

(x) 4x x2

1 4,

(x) 3

x

1 x2

1.

	4.11. (x) esin x 1,		(x)		ln(1 x 5x2)	,	(x) 72x
					1 x 5x2
								x
	4.12. (x) 41 3x5		x7 1,		(x) (5x3 1)tg3
										,
								6

	(x) etg5 x 1.
	4.13. (x) 2x3 arctg		,	(x)	ln(1 4x5)	,
		x3					(x)
					2cos2 x

2 3x 1 7.

34x4 1

1.

3

4.14.(x) 1 52arctg(x3 ), (x) (x 2)log2(1 xsin x),

(x) 1 e3x4 x3 .

4.15. (x) 9(arctg4

x)9

1,

(x) 7x 1sin(x3

),

x

x2

sin x

(x) tg e

1 .

3xsin4x

4.16. (x) etg2 (x2 )

1,

(x)

,

(x) 41 7x2 8x3

4.

cos11x

(x)

arctg4(3

)

,

4.17. (x)

1,

x8

1 x3 sin4 x

(x) e(x 1)2 1 e2.

x2 3x 5

52

4.18. (x) 1 e4x4 , (x) (x2 7)tg3arcsin(x6),(x) log3(cos2 x).

4.19. (x) 1 3 1 x3 2x4,

(x) 2sin3 x 1,

(x) log2(cos 2 x).

x)

4.20. (x) 1 23x,

4ln(1

x

(x)

,

(x) arcsin

tg3 x

.

31 3arctg2 x

4

4.21. (x) 3 tg(sin8 x),

ln(1 5x3),

(x)

cos6(

x

)

(x) 8 3arctgx3 1 .

4.22.(x) 4esin(x3 ) 4, (x) 5lg(1 3x10 )arccos3x,

(x) 4x x2.

4.23. (x) 1 e8x4

,

(x)

24sin2 (x6 ) 1,

(x) log5(cos4 x).

4.24. (x) 7

(x)

3tg(arcsin3 x)

,

1 8x2

4x5

1,

(x) 7sin5 x 1.

2x2 3x 4

4.25. (x) 3arcsinx6,

(x) 3

tg(3x9) 1

1,

(x) ln(2 cosx3).

arctg

(x) 3x2 x 1

(x) 5

1.

4.26. (x) 3

x

x

1,

3,

1 x2

4.27.(x)

(x)

4.28.(x)

(x)

1 3 1 tg3(x4), (x) 5arctgx 4 ,

15x x2

e(x 1)2 2 1.

e

e5tg(x3 ) 1, (x) cosx 3 ln(1 23x),

2x3 x2 1 1.

2

4.29.(x) 1 2tg(x2 3x), (x) 3arcsinx7 arccosx7 ,

(x) 16 43x 2.

4.30. (x) 53x2 2x 1 5,	(x)	ln(1 2x 3x2)	,	(x) etgx 1.
		x2 x 1

53

Задание 5.

При x 0 для функции f (x) найдите главную часть вида Axk .

3

5.1.f (x) 6sin3 x 2tgx2x 9(5arcsin(x 2 )_1).

5.2.f (x) 5arctg34x2 2x2 3sin x cos2x.

2

5.3.f (x) 3x2 2x 1 esin3 x 4 5e x arctg3x.

5.4.f (x) 1 etg(x3 ) 4ln(1 x7 ) arcsin3 x.

x3

5.5.f (x) log2(1 tg2 x3)15 71 x ln(1 3x2) tg9x3.

5.6.f (x) 31 arcsin3(x2) 1 2e5x ln(1 9x7) 2xsin17x10.

5.7.f (x) e9x e3x 5(9 3x2 2) arctgx5 .

4 7x

5.8. f (x) 1 5x2 4x3 2 ex2 sin3 3x.

5.9. f (x) 5arctg3 4x 3sin(x2)cos(x4) xtg2 x .

2

5.10.

f (x) earcsin2 (

x) 6 4x x2 1

3 1 x2.

5.11.

f (x) 72x2

3x 1 8 esin x

ln(1 x 5x2)

.

1 x 5x2

5

x

5.12.

f (x) 41 (3x5

x7) 2 etg

x (5x3 1)tg3

.

6

1

ln(1 4x5)

.

4x4 1

1 2x3 arctg

5.13. f (x)

3

x3

3

2cos2 x

5.14.f (x)

5.15.f (x)

5.16.f (x)

5.17.f (x)

5.18.f (x)

(x 2)log2(1 xsin x) ex3 3x4 2 52arctgx3.

7x sin x3x 6(9arctg4x9 1) tg(ex2 sin x 1).

41 7x2 8x3 etg2 x2 5 3x sin4x. cos11x

e(x 1)2 1 e2 1 x3 sin4 x 1 arctg4 3x8 .

x2 3x 5

log3(cos2 x) 1 e4x4 (x2 7)tg3arcsinx6.

54

5.19.f (x)

5.20.f (x)

5.21.f (x)

5.22.f (x)

5.23.f (x)

5.24.f (x)

5.25.f (x)

2sin3 x 2 31 x3 2x4 log2(cos 2 x).

4ln(1 x x) 5arcsintg3 x 23x 1. 31 3arctg2 x

3tg(sin8 x) 4ln(16 5x3) 8 3arctgx3 1 . cos ( x)

3

10

sin x3

(x 2)

2

.

5lg 1

x

arccos3x 4e

log5(cos4

x) e8x4

24sin2

x6 .

7

sin5 x

3tg(arcsin3 x)

1 8x

2

4x

5

2 7

.

2x

2 3x 4

ln(2 cos(x3)) 1 3tg(3x9) 1 3arcsinx6.

5.26. f (x) 3xarctgx 5 3x2 x 1 51 x2.

	(x 1)2	2		1							(arctg	x	)4
5.27. f (x) e			e		1 3 1 tg		3	x	4					.
											1 5x x2

5.28. f (x) 1 2x3 x2 1 2 e5tgx3 ln(1 2arcsin3x) .

2

3 cos x

5.29.f (x) 1 2tg(x2 3x) 4(16 43x 2) 3arcsin(x7 )arccos(x7 ).

5.30.f (x) 53x2 2x 1 6 etgx ln(1 2x 3x2).

Задание 6.

Вычислите lim f (x) и определите, какое из следующих трех утвержде- x 0 g(x)

ний верно при x 0:

1)f (x) и g(x) имеют одинаковый порядок малости;

2)f (x) (g(x));

3)g(x) ( f (x)).

		e7x e			2x
6.1. lim								.
x 0 5arctg3				2x2 3sin		x
			2x
					2

55

	3x2	2x 1	6 e	tgx	ln(1 2x 3x			2	)
6.2. lim	5									.
						x
x 0		5arctg4x 2x2 3sin
					2

6.3. lim

(x 1)2

etgx3

4ln(1

x7 ) arcsin3 x

.

e8x 2x2 3x 4(16 4x

2 2) arctg

x7

x 0

6.4. lim

3x2 2x 1

esin3 x

4 5arctg3x

.

x 0 e(x 1)2 2

e 1 1 3

1 tg3(x4)

(arctg

)4

x

e9x

23x x2

5(9 3x2 2) arctg

6.5. lim

x

5

.

x 0 e(x 1)2 1

e2

1 sin4(x3)

1 arcsin4 3

x9

lim

log2(1 tg2 x3)5 ln(1 3x2 x3)

.

6.6.

sin9x3

x3

x 0

6tg3 x 2sin x2

9(5arctg

1)

x

1 ln(1 9x7 8x9) sin17x10

lim

3

1 arctg2 x3

6.7.

.

log3(cosx3) 2

e4x7

e3x8

x 0

1 7x2

8x3

sin2 x2

5 3arctg4x

2

6.8. lim

4

e

.

x 0 log2(1 sin2 x) ex3 3x4

2 52arctgx3

5

2

x2

7 1 8x2 4x5

2 7sin

(x

) 3tg4

5

6.9. lim

.

1

x 0

2

x3 x2 1 2 etg x3

ln(1 2arcsin3

)

x

2

x2

4

5

x

1 3x5 x6 2 esin

5tg4

6.10. lim

3

.

x 0

2ln(cosx5) 24sin2(x6 )

e8x13

5arcsin3

2x5

3tg

x

4x2

6.11. lim

3

.

2

x 0 72x

3x 1 8 esin x ln(1 x 8x4)

2 esin x3

0,5 2

x3 x2 1

ln(1 2arctg3

)

6.12. lim

x

.

)4 3

e 1 1 e(x 1)2

2

x 0

(arctg

1 tg3 x4

x

6.13. lim ln(2 cos(sin2 x3)) 3tg(3x9) 1 arcsinx7 1. x 0 31 arctg3(x2) 1 sin3x11 log2(1 5x12)

56

arcsin2

x2 x 1

5

x

5 3

1 x

2

4x

4

6.14. lim

3

.

x 0 2sin x3

2 3 1 x3

5x5

log2(cos 2 x)

e(x 1)2 1

e2

1 arctg4 3

6.15. lim

1 sin4

x3

x8

.

3

10

4

sin x3

2

x 0

5ln(1

) arctg

x 4e

(x 2)

x

(4ln(1

x) 5arcsin

3

(23x 1)2

6.16. lim

x

x

.

etgx3

x 0

(x 1)2

4ln(1

x9 ) arctg3 x

2 ex5 sin3 3x arctg(x

4)

6.17. lim

1 5x2

4x3

.

(72x2 3x 1

8 esin x

ln(1 x 5x2))2

x 0

3x

5arcsin3

4x 3tg(x3) 8tg3

2

6.18. lim

.

x) arctg5

x 0 ln(1

x

x 27x 1

earcsin2

6 4x x2

1

3

x

1 x2 3x5

6.19. lim

.

3tg3 x 2,5sin x2

8(3arctg x3

x 0

x

1)

7sinx3

6(9tg4

x9

1) arctg(ex7

1)

6.20. lim

x

.

log2(1 sin2

x) ex3 3x4

2 52arctgx3

x 0

1

1 2arcsin4

ln(1 3x8

x4 2x5 1

4x5)

3

x3

3

6.21. lim

.

(e(x 1)2

2 e 1

)4)4

x 0

1 3

1 tg3(x4)

(arctg

x

3x2 2x

esin3 x

6.22. lim

e

x

5arctg3x 3

.

x 0 4ln(1 2

x x) 8arcsin3(x2) 25x

1

5

x

x

4 1 x 8x2

2 etg

5sin5

6.23. lim

4

.

x 0 52x2 3x 1 6 etg x

ln(1 x 5x2)

6.24. lim

ln(2 cos(sin x2)) 1 5

sin(5x9) 1

arctg(x7)

.

(3x2 1 esin3 x

4 5tg(x5))2

x 0

3

2

x

6 1 8x2

(x

4x5

2 3tg

)

3sin5

6.25. lim

4

.

ln(2 cos(tgx)) 1 ex3

x 0

arcsin2(x3)

57

	log5		(1 sin	2	x) e	4x	4		3arcsinx5
6.26. lim							2 3			.
					2 esin3x3
x 0		1	6x2 x3					arctg(x4)

		(x 1)2 etg3x 4ln(1 3 x7 ) arcsin							5 x
6.27. lim											.
x 0		earctgx 6 4x x2		1 3 1 x2 3x5
		2log6(cosx) e3x4				25tg2 (x3)
6.28. lim								.
x 0 2 cosx 4 1 x7 4x3						sin3(x2)

6.29. lim

e(x 2)2

3 e 1 5 1 sin2 x3 arctgx

.

53x2 2x 1 6 etgx ln(1 2x 3x2)

x 0

6.30. lim

2log3(cosx4) e5x10

32sin2 (x5)

.

x9

x 0

3

5

x

1 x4

2x5 2 esin

arcsin4

3

Задание 7.

Вычислите пределы:

8

7.1. lim (sin x)tg2 3x.

	x
	2						x
				x		cosec
	7.3. lim(sin				)	2 .
				4
	x 2					1

7.5. lim(cos4 x)sin x.

	x 1					2
7.7.								x.
	lim (cosx) sin2
	x 2
					5
7.9.	lim (cosx)tg5x sin2x.
	x 4
		x	6tg		2	3 x
								2
7.11. lim(sin			)								.
		2
	x 1

7.2. lim (sin2x)cos2x.

x

4

7.4. lim (sin x)cosec2x.

x 2

7.6. lim(sin x)cosec2 x.

x 1 2

4

7.8. lim (cosx)sin3x.

x 2

tgx tg3x

7.10. lim (sin5x)			5	.
x
2			ctg2x

7.12. lim (cos4x)sin 4x.

x

58

1

7.13. lim(cos x)xsin x.

x 2

					1
				x
						cos x
	7.15. lim		tg				.
				2
	x
	2

7.17. lim(cos2 x)ctg x.

x 1

			x		4
						3x
	7.14. lim	cos		sin			.
			2
	x 4				2

7.16. lim (cos2x)cosec4x.

x

7.18. lim (cos4 x)ctg2 x.

x 1

7.19. lim (cos4x)tgx.

x

2

1

7.21. lim (cos2x)sin2 x.

x

3ctgx

7.23. lim (cosx)sin 4x.

x 4

ctg2x

7.25. lim (cos3x)sin5x.

x 2

1

7.27. lim (1 cos3x)cosx.

x

2

5

7.29. lim(cos2x)xsin 4x.

x

Задание 8.

			x ctg2 x
	7.20. lim	sin			.
			4
	x 2

7.22. lim (cosx)ctg2 2x.

x 2

7.24. lim (sinx)6tgx tg5x.

x

2

7.26. lim (sinx)tgx.

x
2									3
					2cos2		x				cos x
7.28. lim													.
							2
x
2
									1
						cos(		3		x)
												.
7.30. lim (tgx)									4

x

4

При каком значении a функция							f (x) будет непрерывной в точке x0, ес-
ли в некоторой окрестности этой точки							f (x) задана следующим образом:
	x2		1		x 1,
8.1.				,		x0 1.
	f (x) lnx
			a,		x 1,

59

8.2.f (x)

8.3.f (x)

8.4.f (x)

8.5.f (x)

8.6.f (x)

1 sin2x							,				x										,
		4x															4					x						.
																							0
																										4
			a,								x
																			,
																4
sin7x sin3x													,						x 2 ,
		ex		2	e4				2																		x0 2 .
			a,																x 2 ,
			x2 x 1							1
															, x 1,											x0 1.
					lnx
			a,																			x 1,
		ln(5 2x)										,				x 2,
																									x0 2.
		10 3x 2
			a,													x 2,
1 2cosx												x
								,														,
			3x														3					x							.
																								0
																										3
			a,									x										,
																	3

3

x 1

,

x 1,

x0

1.

8.7. f (x)

4

x 1

a,

x 1,

1

cosx

,

x 0,

x0 0.

8.8. f (x) 1 cos

x

a,

x 0,

lgx 1

,

x 10,

x0 10.

x 9

8.9. f (x)

1

a,

x 10,

x

cos

2

8.10. f (x)

,

x 1,

x0 1.

1

x

x 1,

,

arcsin

x 2

2

x 2,

8.11. f (x)

,

x0 2.

2 x x

2

3

9

a,

x 2,

60