Zad 1 Korzystając z twierdzenia de l’Hospitala

obliczyć granicę:

a) lim

x→0

arcsin

x
2

sin 3x

b) lim

n→∞

ln(2x+1)

ln(x

2

+2

)

c) lim

x→1

x

3

−1

x

2

−4x+3

d) lim

x→∞

(x − ln x)

e) lim

x→∞

ln

2

x

√

x

f ) lim

x→0

+

ln x · ln (x + 1)

g) lim

x→1

x ln x

x

2

−4x+3

h) lim

x→1

x

10

−1

x

3

−1

i) lim

x→π

sin3x
sin5x

j) lim

x→0

+

ctgx

lnx

k) lim

x→∞

ln x−2x

ln x+x

l) lim

x→∞

ln(x

2

+x)

ln(4x+1)

Zad 2 Zbadać monotoniczność:

a)f (x) = (x

3

− 3x

2

) e

x

b)f (x) =

x

3

1−x

2

c)f (x) = 2x

3

− 9x

2

+ 12x

d)f (x) =

lnx

x

e)f (x) =

x

3

3

+

x

2

2

− 2x + 1 f )f (x) = x

3

e

x

g)f (x) = (x + 1)

2

(x − 2)

h)f (x) = x

4

· ln

3

x

i)f (x) = 3x

5

− 5x

3

j)f (x) = x

2

− lnx

k)f (x) = x

2

√

1 − x

2

l)f (x) = e

2x−x

2

m)f (x) = 2x − ln(2x + 3) n)f (x) = xln

2

x

o)f (x) = (x + 1)

3

· x

5

p)f (x) = x

3

+

3

x

q)f (x) = ln

5

x − ln

2

x

r)f (x) = x · ln

5

x

s)f (x) = x

6

− 6x

4

t)f (x) = x · ln

2

x

Zad 3 Znaleźć ekstrema lokalne funkcji:

a)f (x) = x

5

(x − 2)

5

b)f (x) = 2x +

3

e

x

−1

c)f (x) =

2x

2

+10x+13

xe

2

d)f (x) = x

3

· e

−x

e)f (x) = x

3

+ 3x

2

+ 4

f )f (x) =

lnx

x

g)f (x) = 2sinx + cos

2

x

h)f (x) =

lnx

(x−2)

4

3

i)f (x) = x

3

− 2x

2

+ x

j)f (x) =

2x

2

−1

x+1

k)f (x) =

x

3

3

− 2x

2

+ 3x

l)f (x) =

√

x

2

− 25

m)f (x) =

2x

2

−x+1

x+3

n)f (x) =

2x

x

2

+1

o)f (x) = 2 − (x − 1)

4
3

p)f (x) =

xlnx

1+2lnx

q)f (x) = (x

3

− 3x

2

)e

x

r)f (x) = e

2x−x

2

s)f (x) =

(x−2)

2

2x

t)f (x) =

√

x · e

x

u)f (x) =

sin

2

x

x

w)f (x) = x

3

− 2x + x

x)f (x) =

−π

2

x

y)f (x) =

x

2

2

e

x

z)f (x) =

1

(2x−1)

3

Zad 4 Największa i najmniejsza wartość

a)f (x) = x

4

− 2x

2

w [−2, 2]

bf (x) =)e

−x

− e

−2x

w [−1, 1]

c)f (x) = 4 ln x − x

w [1, e

2

]

d)f (x) = x

2

e

−5x

w [−1, 1]

e)f (x) = x − ln x

w [

1
e

, e]

f )f (x) = x +

√

x

w [0, 1]

Zad 5 Zbadać przedziały wklęsłości i wypukłości

funkcji oraz znaleźć jej punkty przegięcia:

a)f (x) = x

2

− 2x + 2x lnx

b)f (x) =

3

√

1 − x

2

c)f (x) = e

arctg

4

√

x

d)f (x) = arcsin(x

2

)

e)f (x) = ln(ln(x

2

+ 1))

f )f (x) =

sinx

x

2

g)f (x) = x

2

− ln x

h)f (x) = x

2

ln x

i)f (x) = 4x

2

+ x

−1

j)f (x) = ln(tg)

x
3

k)f (x) = x

2

− 2x + 2x ln x

l)f (x) =

3

√

sin2x

m)f (x) = x

4

− 6x

3

+ 12x

2

− 7x + 2

Zad 6 Zbadać przebieg zmienności i narysować

wykres funkcji:

a)f (x) =

x

lnx

b)f (x) =

lnx

√

x

c)f (x) =

1

1+x

2

d)f (x) = −x

3

+ 4x − 3

e)f (x) = sin x − sin

2

x

f )f (x) =

ln x

√

x

Przygotował: Andrzej Musielak