Правило Лопиталя

1.

lim

x2

4x 1

x 4

x 4

2.

lim

x2

2x 1

x 1

x 1

3.

lim

6x2 x 1

x

1

x 12

2

4.

lim

x3

3x 2

x 2

x 2

5.

lim

x2

2x 1

2

x 1

x 1 2x

6.

lim

x3

3x 2

x 2

x

7.

lim

x2

2x 1

2

x 1

x 2x

8.

lim x

x2 4

x

9. lim

6x 3

3

2

4

x 2

x

sin

x

10.

lim

4

x

x 0

11.

lim

tgx sin x

x3

x 0

12.

lim

1

7

x

3x

x

13.

lim

2x 1

4 x

2x 3

x

14.

lim

sin 2 x

x

2

x 0

15.

lim

ln(1 sin 2 x2 )

x2 x4

x 0

1.

lim

x2

9

x 2

x 3

2.

lim

x2

9

x 3

x 3

2x

1

3.

lim

1

x

x 0

4.

lim

x2

2x 1

3

x

2

x 1

x 1 x

5.

lim

(x2

2x 3)2

3

4x

2

3x

x 3 x

6.

lim

x3

12 x 1

4

x

2

1

x 2x

7.

lim

x3 x2

11

3

4x

1

x 3x

8.

lim

x2

x

x2 4

x

9. lim

x3

x

10 x 1

3

x 1

10.

lim

1 cos 5x

x 0

x2

11.

lim 2xsin

4

x

x 0

12.

lim

1

4

2 x 1

5x

x

13.

lim

2x

3

3x 1

2x

5

x

14.

lim

sin 3 2x

4

x 0

arctg

x

3

15.

lim

ln(1 3sin x)

ln(1 3cos x)

x 0

1.

lim

x2 14 x 1

x 3

x 2

2.

lim

x 1

x 2

x 2

2

1

3.

lim 2x

4x

1

x

x 0

4.

lim

x2

2x 1

4

x

2

1

x 1 2x

5.

lim

2x x3

1

x

2

x

x 0

6.

lim

x3

12 x 1

4

11x 2

x 3x

7.

lim

(1 3x)2

2x

2

1

x

8.

lim

x2

7x

x2 1

x

9. lim

x 1

2

x 5

x 5

10.

lim

tg 4x

x 0 sin 5x

11.

lim

2x

x 0 sin 3x

7

3x 1

12.

lim 1

2x

x

2x

4

3x 4

13.

lim

2x

3

x

x 0

sin

x

15.

lim

arcsin

2

x

ln

3

(1 x

2

)

x 0

1.

lim

2x

4

2

2

x 1

x

2.

lim

x2

x 1

x 1

x 1

3.

lim

4x 1

1

x

x 0

4.

lim

x3 4x2 5x 2

x

3

3x 2

x 1

5.

lim

x3 3x 2

x

2

x 2

x 1

6.

lim

x3

3x 2

2x

2

1

x

7.

lim

2x2 4x 1

4x4 1

x

8.

lim

3x2

1

3x2 x

x

9. lim

x2

1 1

x

x 0

10.

lim

tg12 x

x 0

x

11.

lim

cos 12 x cos 3x

x 0

x2

12.

lim

1

4

2 x 1

3x

x

13.

lim

3x 4

8 x 1

3x 1

x

14.

lim

ln(1 xsin x)

tg 2 x

x 0

15.

lim

3arcsin

x3

tg 3 x

x 0

1.

lim (x2

1

2

)

2

x 0

x

2.

lim

x2

1

x 1

x 4

3.

lim

x2

1

x 2

x 2

4.

lim

x3 5x2 7x 3

3

4x

2

5x 2

x 1 x

5.

lim

(x2

2x 3)2

3

4x

2

3x

x 3 x

6.

lim

x4

11x2

1

2x

3

x 1

x

7.

lim

x3

11x 2

2

3x

3

1

x 4x

9.

lim

2

x 1

2

25

x 5

x

10.

lim x ctg 7x

x 0

11.

lim

sin 4x sin x

x 0

x

12.

lim

1

4

2 x

3x

x

13.

lim

4x 2

4 x 3

4x 3

x

14.

lim

sin 3 4x

3

(1 2x

2

)

x 0 ln

15.

lim

sin 4x2

x 0 sin 15x2

1.

lim (3x2

4x 8)

x 0

2.

lim

x2

16

x

2

16

x 4

3.

lim

4x2

14 x 6

x 3

x 3

4.

lim

x2

2x 1

x

3

x

2

x 1

x 1

5.

lim

(2x2 x 1)2

3

2x

2

x 2

x 1 x

6.

lim

x2

2x 1

3

x

2

1

x x

7.

lim

4x2

12

x

2

4x

x

8.

lim

x2

3x x2

1

x

9. lim

x2

16

4

3x 4

x 5

10.

lim x ctg 3x

x 0

sin

x

11.

lim

3

4x

x 0

12.

lim

1

4

x 1

3x

x

13.

lim

3x 3

4 x

3x 4

x

14.

lim

12 x2

x 0 sin x2

15.

lim

ln(1 sin 2 x)

x2 cos 2

x

x 0

1.

lim

x2 16

x 2

x 4

2.

lim

x2

2x 1

x 1

x 1

3.

lim

3x2

5x 2

x 2

x 2

4.

lim

x4 1

4

x

2

1

x 1 2x

5.

lim

x3

3x 2

x x

2

x 1

6.

lim

x2

x 1

3

x

2

2

x x

7.

lim

4x4

x2

1

x

4

x

3

1

x

9.

lim

x2

x

9x 7

x 1 4

10.

lim

1 cos 8x

1 cos 3x

x 0

11.

lim

sin 4

x

x

4

x 0

3 x 2

12.

lim

1

x

x

13.

lim

2x

1

2 x

x 2x

1

14.

lim

ln(1 8x2 )

sin 3x

x 0

15.

lim

sin 6x

4x x3

x 0

1.

lim

x2

4x 1

x 2

x 4

2.

lim

x2

9

x 3

x 3

3.

lim

6x2 x 1

x 1

x 1

2

2

4.

lim

x3

3x 2

x

2

2x 1

x 1

5.

lim

(x3

2x 1)2

x

4

2x 1

x 1

6.

lim

x3

3x 2

2

2x 1

x x

7.

lim

x2

4x 1

x

4

1

x

9. lim

x2

2x

6x 8 2

x 2

sin(

x) sin(

x)

10.

lim

3

3

x

x 0

11.

lim

x sin 4x

2x sin 3x

x 0

12.

lim