Правило Лопиталя

1.

lim

x

2

2

4

x 2

x

2.

lim

x 1

x

2

4x 1

x 1

3.

lim

x2

4x 3

x 3

x 3

4.

lim

x3

5x2 8x 4

x

3

3x

2

4

x 2

5.

lim

(x2

2x 3)2

3

4x

2

3x

x 3 x

6.

lim

x2

12 x 1

x

3

x

2

x

x

7.

lim

x3

4x2

1

3

11x

2

x 8x

9. lim

x3 x

10 x 1

3

x 1

10.

lim

1 cos 4x

x 0

x2

11.

lim 3xsin

5

x

x 0

4

8 x 1

12.

lim

1

3x

x

13.

lim

7x

4

2 x 3

3

x 7x

14.

lim

x2 x

sin 2 x

x 0

15.

lim

ln(1 3xsin x)

ln(1 3xcos x)

x 0

1.

lim

x2

x 2

x 4

x 1

2.

lim

x2

4x 2

x

2

4

x 2

3.

lim

2x2

3x 2

x

1

x 12

2

4.

lim

x4

1

4

x

2

1

x 1 2x

5.

lim

(1 x)3 (1 3x)

x

5

x

x 0

6.

lim

x4

1

4

x

2

1

x 2x

7.

lim

(1 x)3

1)

2

x (x

8.

lim

3x

9x2

1

x

9. lim

x 1

2

x 5

x 5

10.

lim

tg3x

x 0 sin 4x

11.

lim

4x sin 3x

2x sin 5x

x 0

12.

lim

1

7

2 x 1

3x

x

13.

lim

3x 3

3x 3

3x 4

x

14.

lim

ln(1 x2 )

tgx

x 0

15.

lim

arcsin 2

x

3

(1 x)

x 0 ln

1.

lim

x2

4x 5

x 1

x 1

2.

lim

x2

4

x

x 0

3.

lim

6x2

5x 1

x 1

x 13

3

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

x

2

x 2

x

7.

lim

4x3 8x2

1

x

3

7x

2

1

x

8.

lim

x

x2

1

x

9.

lim

x2

1 1

x

x 0

10.

lim

tg8x

x

x 0

11.

lim

cos 5x cos 3x

x 0

x2

12.

lim

1

1

2 x

3x

x

13.

lim

2x 1

4 x 1

2x 1

x

14.

lim

ln(1 x cos x)

tgx

x 0

15.

lim

2arcsin

x

3x

x 0

1.

lim

x2

4

x 1

x 1

2.

lim

x2

4

x 8

x 2

3.

lim

10 x2

9x 7

x 7

x 7

5

5

4.

lim

x3

x2

5x 3

x

3

x

2

x 1

x 1

5.

lim

x3

2x 1

4

2x 1

x 1 x

6.

lim

x3

2x 1

4

2x 1

x x

7.

lim

x3

12 x2

1

4x

3

x

2

1

x

8.

lim

2x

4x2 1

x

9. lim

x 1

1

x

2

x 0

10.

lim

(1 cos x)2

3

x sin

3

x

x 0 tg

11.

lim

tg3x

x 0 tg8x

12.

lim

1

8

3x 1

7x

x

13.

lim

3x 1

2 x 1

3x 4

x

14.

lim

ln(1 x2 )

arcsin

x

x 0

15.

lim

tg 3 x

arctg 3 x

x 0

1.

lim

x2

1

x

2

4

x 1

2.

lim

x2

4

x 2

x 2

3.

lim

2x2

13x 21

2x 7

x 72

4.

lim

x3

3x 2

x

3

x

2

x 1

x 1

5.

lim

x2 1

2

x 1

x 1 2x

6.

lim

x2

11x 1

3

12 x 1

x x

7.

lim

2x2

12 x 1

x 11

x

9. lim

3x 1

3x2

1

x

10.

lim

1 cos x

x 0

x2

11.

lim

tgx sin x

x3

x 0

12.

lim

1

11

3x

7x

x

13.

lim

2x 4

4 x 2

2x 3

x

14.

lim

ln(1 4sin x)

arcsin

x

x 0

15.

lim

sin 6x

x x7

x 0

1.

lim

x

1

2

4

x 1

x

2.

lim

x2

2x 1

x

x 0

3.

lim

2x2

9x 10

2x 5

x 5

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

x3 12 x

x

4

11

x

7.

lim

x4 x3 1

4

x

2

x 11

x 2x

8.

lim

x2

5x 3 x2

3x

x

9.

lim

2

x 3

2

49

x 7 x

10.

lim x ctg 5x

x 0

11.

lim

4x sin x

3x sin 4x

x 0

12.

lim

1

5

4 x

7x

x

13.

lim

3x 4

4 x 3

3x 3

x

14.

lim

ln(1 sin 2 x)

tg 2 x

x 0

15.

lim

sin x cos x

x 0

x x3

1.

lim (x 4x 1)

x 2

2.

lim

x2

4

2x

2

8

x 4

3.

lim

6x2

x 1

x

1

x 1

3

3

4.

lim

x3

3x 2

x

2

x 2

x 1

5.

lim

x3

3x 2

x 2

x 2

6.

lim

x3

3x 2

x

2

2

x

7.

lim

x2

12

x

3

x

x

8.

lim

x2

4x

x2 3

x

9. lim

9 x2

3x 3

x 3

10.

lim

sin x sin 5x

x 0

x

11.

lim

sin 3x

x 0 sin 8x

3

x

12.

lim

1

2x

x

13.

lim

2x

1

2 x

3

x 2x

14.

lim

sin 3 3x

ln

2

(1 2x

2

)

x 0

15.

lim

sin 4x

x 0 sin 8x

1.

lim

x2

4x 1

x

2

1

x 1

2.

lim

x2

4

x 1

x 0

3.

lim

6x2

75 x 39

x

1

x 1

2

2

4.

lim

x2

2x 1

2

x 1

x 1 2x

5.

lim

(x2

3x 2)2

3

2x

2

x 2

x 1 x

6.

lim

x2

2x 1

2

x 1

x 2x

7.

lim

x2 3x 2

3

2x

2

2

x x