Katedra Ekonometrii UŁ

Zestaw nr 3: Granica i pochodna funkcji

Zadanie 1. Oblicz granice funkcji:

(1)

lim

x→∞

2x

2

+ 1

x

2

+ 2x − 1

(2)

lim

x→∞

√

x

2

+ 3

7x − 9

(3)

lim

x→∞

√

x+2+

√

x−5

x

(4)

lim

x→∞

√

x + 2 −

√

x − 5

(5)

lim

x→−∞

√

x

2

− 3 −

√

2x

2

+ 2

(6)

lim

x→−∞

2x+5

2x

7x+2

(7)

lim

x→∞

x−3

x

7x+2

(8)

lim

x→−∞

1+e

x

2x−6

(9)

lim

x→∞

2−e

−x

x

2

+6

(10)

lim

x→−∞

e

1+x

2x−6

(11)

lim

x→∞

xe

1

x

(12)

lim

x→−∞

e

1

x

ln x

2

(13)

lim

x→−∞

7x

3

+2x

2

−7

x

2

+1

(14)

lim

x→∞

4x

2

−5x

2x

4

−1

(15)

lim

x→−∞

sin(x

2

+3x)

3x

10

+13

(16)

lim

x→∞

x

√

x

2

+ x

(17)

lim

x→∞

x

√

2

x

+ 3

x

(18)

lim

x→∞

(

x

√

4 − ln(1 +

1

x

))

Zadanie 2. Oblicz granice funkcji:

(1)

lim

x→0+

4

ln(x+1)

(2)

lim

x→0

−

−2

ln(x+1)

(3)

lim

x→0

x

2

+2x

2x

2

+x

(4)

lim

x→1

x

2

−3x+2

x−1

(5)

lim

x→2

+

x

2

+4

x−2

(6)

lim

x→2

x

2

−4

x−2

(7)

lim

x→−1

+

ln



1

x+1



(8)

lim

x→1

−

ln



−5

x

2

−1



(9)

lim

x→3

+

e

1+x

x2 −4x+3

(10)

lim

x→3

−

e

1+x

x2 −4x+3

(11)

lim

x→1

+

1

(x−1)

2

(12)

lim

x→1

−

1

x−1

(13)

lim

x→∞

e

√

x+1

e

√

x

(14)

lim

x→1

1

√

x−1

(15)

lim

x→−∞

7x

3

+2x

2

−7

x

2

+1

Zadanie 3. Oblicz granice funkcji wykorzystując regułę de l’ Hospitala :

(1)

lim

x→0

4x

sin 3x

(2)

lim

x→0

−

e

1

x

ln(x+1)

(3)

lim

x→1

tg(x−1)

4x−4

(4)

lim

x→2

x

5

−32

x−2

(5)

lim

x→0

+

xe

1

x

(6)

lim

x→0

e

x

−e

−x

sin x

(7)

lim

x→∞

2x

2

−5

e

x

(8)

lim

x→1

2−2x

ln x

(9)

lim

x→0

+

x

2

ln x

(10)

lim

x→0

1−cos

2

x

sin x

(11)

lim

x→∞

ln x

x

2

+1

(12)

lim

x→−∞

e

−x

x

3

+1

Zadanie 4. Oblicz pochodne funkcji:

(1)

f (x) = 2x

3

+ 4x

2

− 5x + 4

(2)

f (x) =

7

4

√

x+3

(3)

f (x) = x

2

e

x

(4)

f (x) =

2x − 1

√

2

(5)

f (x) =

x

2

− 1

√

x

(6)

f (x) = 4 cos(3x) − 2 sin

2

x

(7)

f (x) =

e

3x

x

2

+4

(8)

f (x) =

4x

5+2x

(9)

f (x) =

4

x

3

−

3

x

2

+

5

x

(10)

f (x) = e

7+3 ln x

(11)

f (x) = sin 4x

2

+ 11

(12)

f (x) = e

2x cos x

(13)

f (x) =

q

1−2x

1−x

(14)

f (x) = ln

5

x−2

(15)

f (x) = ln (sin 3x)

(16)

f (x) = x

3

ln(5x)

(17)

f (x) = ln(x

2

− 1)

(18)

f (x) = ln

e

x

sin x

(19)

f (x) = x

3

ln

2

(x)

(20)

f (x) = e

x

ln

2

(x)

(21)

f (x) = sin

2

x + cos

2

x

1

Katedra Ekonometrii UŁ

Odpowiedzi

Zadanie 1.

(1)

2

(2)

1
7

(3)

0

(4)

0

(5)

−∞

(6)

e

35

2

(7)

e

−21

(8)

0

(9)

0

(10)

e

1
2

(11)

∞

(12)

∞

(13)

−∞

(14)

0

(15)

0

(10)

1

(17)

3

(18)

1

Zadanie 2.

(1)

∞

(2)

∞

(3)

2

(4)

−1

(5)

∞

(6)

4

(7)

∞

(8)

∞

(9)

∞

(10)

0

(11) ∞

(12) −∞

(13) 1

(14) nie istnieje

(15) −∞

Zadanie 3.

(1)

4
3

(2)

0

(3)

1
4

(4)

80

(5)

∞

(6)

2

(7)

0

(8)

2

(9)

0

(10)

0

(11)

0

(12)

−∞

Zadanie 4.

(1)

f

0

(x) = 8x + 6x

2

− 5

(2)

f

0

(x) = −

7
4

x

0

(x+3)

4

√

x+3

(3)

f

0

(x) = 2xe

x

+ x

2

e

x

(4)

f

0

(x) =

√

2

(5)

f

0

(x) = 2

√

x −

1

2x

3
2

x

2

− 1

(6)

f

0

(x) = −4 cos x sin x − 12 sin 3x

(7)

f

0

(x) =

(

3x

2

−2x+12

)

e

3x

(x

2

+4)

2

(8)

f

0

(x) = 20 (2x + 5)

−2

(9)

f

0

(x) =

6

x

3

−

5

x

2

−

12
x

4

(10)

f

0

(x) =

3

x

e

3 ln x+7

= 3e

7

x

2

(11)

f

0

(x) = 8x cos 4x

2

+ 11

(12)

f

0

(x) = e

2x cos x

(2 cos x − 2x sin x)

(13)

f (x) = −

1
2

1

(x−1)

2

√

1

1−x

(1−2x)

(14)

f

0

(x) = −

5

(x−2)

2

1
5

x −

2
5

(15)

f (x) = 3

cos 3x

sin 3x

(16)

f

0

(x) = x

2

+ 3x

2

ln 5x

(17)

f

0

(x) = 2

x

x

2

−1

(18)

f

0

(x) = 1 −

cos x

sin x

(20)

f

0

(x) = 2x

2

ln x + 3x

2

ln

2

x

(20)

f

0

(x) = (ln

2

x)e

x

+

2

x

(ln x) e

x

(21)

f

0

(x) = 0

2