ТР№2 Пределы

1.

lim f (x) = +∞

x→4

2.

lim x3 = −1

x→ −1

1

8

3.

lim

−

4

16

− x2

x→ −4 x +

4.

x

2

+5x + 4 −

x

2

lim

+ x

x→+∞

5.

lim

1 −2cos x

x→π

3

π−3x

x+2

−e

x2 −4

lim

tg e

6.

ln(3x +7)

x →−2

1

7.

lim[1 − xsin2 x]ln(1+πx3 )

x→0

8.

а) f (x) = x2 +6x +3 x ,

1

1

б) f2 (x) =

,

x0 = ∞

(2x +1) sin2

1

x

9.

а) f (x) = 2x +3arcsin2 x −3arctg 4x ,

x = 0

1

1

0

б)

f2

(x)

=

,

x0 =1

sin πx tg 3πx

в) f3 (x) = x ( x + 2 + x −3), x0 = ∞

10.

f1(x) =

|

x |,

x ≤ 2

f2

(x) =

1

а)

x >

б)

−2),

2

− x)2

lg(x

(1

x→2

3.

lim

x3 −3x −2

x→ −1 x3 + 2x2 − x −2

4.

lim(x +3 1 − x3 )

x→∞

5.

lim

1 −2cos x

sin(π−3x)

x→π

3

6.

lim

esin 2x −etg 2x

x→π

ln

2х

2

π

7.

x2 +5 4x2

lim 2

x→∞ x

+3

8.

а) f (x)

5

−x2

−

=

1 x ,

1

б)

f2 (x) = (1 −cos 6x) tg 2x ,

x0 = 0

9.

а) f (x) = 3

x2 −23 x +1,

x =1 ,

1

0

б) f2 (x) = (x2 +5x)2 tg

1

,

x0 = ∞

x +1

(x) = ex2

в)

f

3

+e−3x

x + 2sin2 x −2 ,

x = 0

0

4x

, x <1

1

10.

а)

f

(x) = 5 − x2 ,

1

< x

≤ 4

б) f

2

(x) = arctg

1

lg(x −4),

x > 4

x +2

x→+0

3.

lim

x4 + 2x3 + x2

x4 + 2x +1

x→ −1

3

4.

lim x

(

x3 + 2 −

x3 −2)

2

x→+∞

5.

lim

(2x −1)2

esin πx −e−sin 3πx

x→

1

2

6.

lim

ln(1 + x 1 + xex )

x→0

cos x −1

5

lim(cos x)

7.

tg 5x sin 2x

x→4π

8.

а) f (x) = (2 − x)4

−(3 + x)4 ,

1

(x3 +3x2 )

б)

f

2

(x) =

, x = ∞

sin

1

0

x

9.

а) f (x) = 4 x4 + x2 + x3 , x = ∞ ,

1

0

б)

f

2

(x) = ln(1 + 2sin 2x + tg2

x ) ,

x

= 0

0

в)

f

3

(x) = (x3 −1)2 3 x −1 ,

x =1

0

−1,

x ≤ −π/ 2

sin 4x

10. а)

f

(x) =

tg x,

−π/ 2 < x < 0

б)

f

2

(x) =

1

| x |

x

> 0

x,

1.

lim f (x) = −∞

x→4−0

2.

lim4−x = 0

x→+∞

3.

lim

(1 + x)3 −(1 +3x)

x→0

x2 + x5

3

2

4.

lim

−

1 −

x

−3 x

x→1

1

5.

lim

sin 4x −2sin 2x

x→0

x ln cos 6x

6.

lim

2x −

3x2 −5x + 2

arctg

х−2

x→2

2

2

lim(2 −3arctg2

7.

x )sin x

x→0

8.

а)

f (x) = sin10x −4sin x2 ,

1

(x) = e6x2 −1,

б)

f

2

x = 0

0

9.

а)

f (x) = (x + 2)(ex2 −5 −e−1) , x

= −2 ,

1

0

б)

f

2

(x) =1 −sin πx , x =1

2

0

в) f3 (x) = 2x2 −4 3 x12 −5x3 +1 ,

x0 = ∞

2

x