1. ] lim lim lim ¨¸ f f ®¾ - МАТЕМАТИКА

1.

a) lim x0

sin 3x 2x

0 0

? LP

lim x0

sin 3x ' 2x '

lim x0

3 cos 2

3x

3cos0 3 1 3

2

22

) lim

x0

ex sin

1 2x

e0 1

sin 0

1 0

1

0 0

?

LP

lim x0

ex 1 '

sin 2x '

lim x0

ex 2 cos

2x

e0 2 cos

0

1 2

) lim

x0

ex

x

ex sin

x

2x

1-1-0

0-0

0 0

?

LP

lim x0

ex ex 2x

x sin x'

'

lim x0

ex ex 1 cos x

2

1 1-2

1-1

0 0

?

LP

lim x0

ex ex 2

1 cos x '

'

lim x0

ex ex sin x

1

0

1

0 0

?

LP

lim x0

ex ex

sin x'

'

lim x0

ex ex cos x

11 2 1

) xlim1

x

x

1

1 ln x

?

x ln x x 1

lim x1

x

1ln x

1 0-1 1

1

1

0

0 0

? LP

x ln x x 1 '

lim x 1

x

1ln x'

ln x x 1 1

lim x1ln

x

x

x

1

1

ln x 1 1

lim x1ln

x

1

1

ln x lim x1ln x 1

1

0

0 1-1

0 0

? LP

x

x

x

1

1

ln x'

= lim x1ln x

1

1 x

'

lim x1

1 x

x

1 x2

1

1

1

11

1 2

) lim

x1

2 x2 1

x

1 1

?

2 x 1 xlim1x 1 x 1

lim x1

x

1 x

1 x

1

0 0

?

xlim1

x

x 1

1 x

1

lim x1

1 x 1

1 2

) lim x1

x4 x3

x3 x2

x 1 x 1

1-1-1 1 1 1 1 1

0 0

? LP

lim x1

x4 x3

x3 x2

x 1 x 1

' '

4x3 3x2 1

lim x1

3x2

2x

1

4-3-1

3-2-1

0 0

?

LP

lim x1

4x3 3x2 1 '

3x2 2x 1 '

12x2 6x

lim x1

6x 2

12-6 6 3 6-2 4 2

) lim x0

cos 2x

2 cos x2

x

1

1-2

0

1

0 0

? LP

lim x0

cos

2x

2 cos x2 '

x

1 '

2 sin 2x 2 sin x

lim x0

2x

0 0

?

LP

4 cos 2x 2 cos x

lim x0

2

-4 2 -1 2

) lim x0

ex sin x ln(1 x)

1

1 0-1

ln1

0 0

?

LP

lim x0

ex

cos x 1

1 1 1

2 1

2

1 x 1 0

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