Правило Лопиталя

1.

lim

x2

4x

x

5

x 5

2.

lim

x 1

x

x 0

3.

lim

5x2 24 x 5

x 5

x 5

4.

lim

x3

2x 1

x

4

2x 1

x 1

5.

lim

(x2

2x 1)2

x

4

2x 1

x 1

6.

lim

(x 1)( x2 4x 1)

x

2

4

x

7.

lim

x 12

2

4x 2

x x

8.

lim

x2

4x 1 x2

3x 1

x

9. lim

x 1 2

x 5

x 5

11.

lim

1 cos 2x

x

x 0

12.

lim

1

4

2 x

5x

x

13.

lim

5x 3

4 x

5x 2

x

14.

lim

sin 2 x

x

x 0

15.

lim

ln(1 3sin x)

arctgx

x 0

1.

lim

x2

4

x 1

x 1

2.

lim

x2

x

x 1

x 0

3.

lim

2x2

6x 8

x 4

x 4

4.

lim

x3

3x 2

x 2

x 2

5.

lim

(2x2

x 1)2

3

2x

2

x 2

x 1 x

6.

lim

x4

12 x 1

x

3

4x

x

7.

lim

x2

15 x 4

4x

2

4

x

9. lim

x 2

1

x

2

9

x 3

10.

lim

x sin 3x

1

x sin 4x

x 0

2

11.

lim

x sin

2

x

x 0

12.

lim

1

3

2 x

2x

x

13.

lim

2x 3 x

2x 1

x

x 0

tgx

15.

lim

ln(1 sin 2 x)

x2 x3

x 0

1.

lim

x2

4x 8

x

2

x 2

2.

lim

x2

8

x 2

x 1

3.

lim

3x2

40 x 128

x 8

x 8

4.

lim

(x3

2x 1)( x 1)

x

4

4x

2

5

x 1

5.

lim

x3

3x 2

x 2

x 2

6.

lim

2x2

5x 3

x 3

x

7.

lim

6x3

4x 1

x

3

x

x

9. lim

1 x2 1

x

x 0

10.

lim

1 cos x

1 cos 3x

x 0

11.

lim

sin 2 8x

x2

x 0

12.

lim

1

7

x

3x

x

13.

lim

3x 1

6 x 1

3x 4

x

14.

lim

x2 x

sin 8x

x 0

15.

lim

ln(1 tg 2 x)

sin 4x

x 0

1.

lim

x2

4

x

2

x 2

2.

lim

x2

4

x

2

1

x 1

3.

lim

6x2

x 1

3x 1

x 13

4.

lim

x3

6x2 12 x 8

x

3

3x

2

4

x 2

5.

lim

(x2

3x 2)2

3

2x

2

x 2

x 1 x

6.

lim

7x2

8x 1

3x 1

x

7.

lim

3x3

4x

x

4

12

x

9. lim

x 1

1

x

2

4

x 2

sin(

x) sin(

x)

10.

lim

2

2

x

x 0

11.

lim

1 cos 3x

x 0

x2

14

2 x

12.

lim

1

3x

x

13.

lim

3x

4

2 x 2

3x 2

x

14.

lim

ln(1 3x2 )

tgx 2

x 0

15.

lim

ln(1 3x)

x 0

sin 4x

1.

lim

x2

1

x 4

x 0

2.

lim

x2

4

x 2

x 2

3.

lim

7x2

8x 1

x 1

x 1

4.

lim

x3

5x2 8x 4

x

3

3x

2

4

x 2

5.

lim

(x2

3x 2)2

3

2x

2

x 2

x 1 x

6.

lim

x3

5x2 8x 4

x

3

3x

2

4

x

7.

lim

x4

3x2

4

x

3

x 2

x

8.

lim

x4

2x

x4 x

x

9. lim

3x 5

2

x

2

9

x 3

10.

lim

cos 3x cos 5x

x 0

x

2

11.

lim

4x sin 3x

3x sin x

x 0

12.

lim

1

3

x

2x

x

13.

lim

3x 4

3x 1

3x 5

x

14.

lim

sin 2

x

tg

2

x

x 0

15.

lim

ln(1 x2 )

sin 2 x

x 0

1.

lim

x2

4

x

2

4

x 2

2.

lim

x2

4

x

2

4

x 2

3.

lim

3x2

2x 1

x 1

x 13

3

4.

lim

x3

6x2 12 x 8

x

3

3x

2

4

x 2

5.

lim

(x3

2x 1)( x 1)

x

4

4x

2

5

x 1

6.

lim

x3

4x 1

x

2

12

x

7.

lim

x4

2x 1

4x

4

x

2

x

8.

lim

x2

2x

x2 2

x

9. lim

x2

7x

5x 1

1

x 0

sin

x

10.

lim

4

3x

x 0

11.

lim x ctg 8x

x 0

12.

lim

1

3

4 x

2x

x

13.

lim

2x

3

2 x 3

2x 4

x

14.

lim

tg 2 x

x

2

x

x 0

15.

lim

ln(1 x2 sin x)

x2

x

x 0

1.

lim

x2

4x 1

x 1

x 1

2.

lim

x2

4x 8

x

2

4

x 2

3.

lim

2x2

5x 3

x 3

x 3

4.

lim

2x2 x 1

3

2x

2

x 2

x 1 x

5.

lim

x3 7x2 15x 9

3

8x

2

21x 18

x 3 x

6.

lim

2x2

5x 3

x 3

x

7.

lim

4x3

12 x 3

2x

3

x

2

1

x

9.

lim

x3 4x

4x 1

3

x 2

10.

lim

x sin 7x

sin 3x

x 0 2x

sin 2

x

11.

lim

3

x

x 0

12.

lim

1

3

x

2x

x

13.

lim

x 3

2 x

x 1

x

14.

lim

x2 x3

sin

2

x

x 0

15.

lim

ln(1 tg 2 x)

sin

2

x

x 0

1.

lim

x2

9x 1

x

2

9

x 3

2.

lim

x2

4x 5

x 1

x 1

3.

lim

5x2 4x 1

x 1

x 1

4.

lim

x2

3x 2

3

2x

2

x 2

x 1 x

5.

lim

x3 4x2 3x 18

3

5x

2

3x 9

x 3 x

6.

lim

x2

3x 2

3

2x

2

x 2

x x

7.

lim

x3

12 x 1

x

2

x

1

x

9. lim

7x 2 3

x

3

x

x 1

10.

lim

sin 8

2x

x8

x 0

11.

lim x ctg 3x

x 0