zadzarz.pdf

Zestaw I - Rownania i nierownosci kwadratowe, logarytmiczne i wykladnicze.

1. Rozwiazac rownania:

a)−x 2 −x+ 6 = 0 b)x 2 + 3x−4 = 0 c)x 2 −2x+ 6 = 0

d)−x 2 −4x−1 = 0 e)2x 3 +x 2 −13x+ 6 = 0

f)x 4 −4x 3 +x 2 −4x= 0 g)2 2x −2 x = 0

h)4 x −3(2) x −4 = 0 i)4 2x + 3(4) x + 2 = 0

j)3 2x+1 −4(3) x + 1 = 0 k)x(2) x −2 x+1 = 0

l)log 2 (x−4) = 0 m)log 2 (8−2x)−2 log 2 (2−x) = 1

n)log 2 (x−4)−2 = 0 o)(log(x)) 3 −4 log(x) = 0

p)(log(x)) 3 + (log(x)) 2 = 0 q)log x (x+ 2) = 2

r)xlog(x) + log(x)−1 =xs) 3

2x−1 = 2

v)2 sin(2x) + p 3 = 0 w)sin(x) = cos(x)

x)(sin(x)−cos(x)) 2 = sin(2x) y)(cos(x)) 4 −(sin(x)) 4 = 1

z)2(cos(x)) 2 + 3 cos(x)−2 = 0

x+2 |>1

f)|x 2 −2x|2−xg)|x−x 2 |+ 40

h)x 4 +x 3 −x 2 +x−2<0 i)2x 4 −x 3 −3x 2 +x+ 1<0

j)x 5 −2x 4 +x−2>0 k)4 x −2 x 0

l)2 −3x <1 m)3 2x+1 + 5(3) x −2>0

n)2

p (x+1) 1 o)2 2x+1 −3(2) x + 10

p)(x+ 1)(2 x −1)<0 q)log(x 2 −3)0

r)2 log 4 (x+ 1)1 s)log 2 (x−1) + log 2 (x+ 1)2

t)log 4 (x+ 1) 2 1 u)log 2 (x 2 −1)2

v)log 1 2 (x+ 2)<log 1 2 (x+ 1 ) w) 2

p (3)

x−2 1

x)sin(x)> 1 2 y)sin(2x)>

3. Wyznaczyc dziedzine fun kcji:

(3 2x −3 x+1 ) b)y= 1− q

(2 x + 3)

c)y= q (1−2 x ) + p xd)y= log (5 x −5 |2x−3| )

e)y= 1 3 x 2 + 5x+ log (3 x 2 −81) f)y= log(x 2 −x+ 2)

g)y= l og(−x) h)y= log(4x−|x 2 −5|) i)y= x

1−log(x)

(log 2 (x+ 2)) k)y= q 4−(log 1 2 (x)) 2

j)y=

t) |x−2|

|x|−2 = 1 u)2 si n(x)−1 = 0

2. Rozwiazac nierownosci:

a)x 2 −4x+ 3<0 b)−3x 2 −21x−30>0

c)|x|> 1 x d)2|x|−|2x+ 3|<1 e)| x

p (x+1) −4

z)4(cos(x)) 2 −3>0

a)y=x− q

l)y=

(2 log(x)−(log(x)) 2 ) m)y= 5 log 3+x (−x−1)

Zestaw II - Ciagi liczbowe.

2 n +5

b)lim n!1 log 2 (n+ 3)

c)lim n!1 log 2 (n+1)

log 3 (n+1)

d)lim n!1 n+3

2n 2 +1

e)lim n!1 4n 3 +6

n+1

f)lim n!1 2n 2 +3n

6n 2 +4n−1

g)lim n!1 (4 n + 2)

h)lim n!1 3 n −2 n

4 n −3 n

i)lim n!1 3 2n+1 −7

9 n +4

j)lim n!1 n5 n

2 n 3 n+1

k)lim n!1 2 n

l)lim n!1 2n!

n n

m)lim n!1 (n 2 +1)(2n−1)!

n)lim n!1 (10− p n)

o)lim n!1 n

(2n+1)!+1

p)lim n!1 ( p n− p n+ 2 )

q)lim n!1 (

p n 2 + 1

n 2 + 4n+ 1− p

n 2 + 2n)

p n 3 + 1

3 p n 5 +1+1

r)lim n!1

2. Korzystajac z twierdzenia o trzech ciagach obliczyc granice:

a)lim n!1 2n+(−1) n

c)lim n!1 n q

4n 2 −cos(n 2 )

d)lim n!1 n+1 p 2n+ 3

3 + sin(n )

e)lim n!1 n p 3 n + 5 n + 7 n

f)lim n!1 n q

g)lim n!1 n q 3 n +2 n

( 2 3 ) n + ( 3 4 ) n

h)lim n!1 n+2 p

5 n +4 n

3 n + 4 n+1

3. Korzystajac z deﬁnicji liczby e obliczyc granice:

a)lim n!1 (1 + 1 n ) 3n−2

1. Oblicz podane granice:

a)lim n!1 1

3n+2

b)lim n!1 2n 2 +sin(n!)

n 2 ) 2n 2 +1

d)lim n!1 ( n 2 −1

n 2 ) 2n 2 −3

e)lim n!1 ( n+4

n 2 ) 2n 2 +1

g)lim n!1 ( n

n+1 ) n

h)lim n!1 ( 3n+1

3n+2 ) 6n

i)lim n!1 ( n 2 +3n+2

n 2 +2n ) 3n+1

Zestaw IV - Granice funkcji:

1. Obliczyc granice:

a)lim x!0 x+4

x−8 ,

c)lim x!0 x 2 −4

x 2 −x−2 ,

d)lim x!2 x−2

x 2 +x−2 ,

e)lim x!2 |x − 4|

p x+2 ,

f)lim x!−1 ( 4x 2 −5x + 7),

g)lim x!0 p 2x 2 + 9,

h)lim x!1 x−1

x 2 +x−2 ,

i)lim x!−1 x+1

x 2 −x−2 ),

j)lim x!2 x 2 −4

x 2 −x−2 ,

k)lim x!2 x 3 −8

x−2 ,

l)lim x!3 x 2 −5x+6

x 2 −8x+15 ,

m)lim x!0 x 3 −4x 2 +5x

(|x|+1)x ,

n)lim x!0 x 3 −4x 2 +5x

4x+x 2 −5x 3 ,

o)lim x!1 2x+2

x 2 −1 ,

p)lim x!−2 (x+2)(3x−1)

4−x 2 ,

x 2 −3x ,

r)lim x!−1 x 2 +5x+4

x 3 +1 ,

s)lim x!0 n4x 2 −3x

7x ,

t)lim x!0 5x 2 −x

2x 2 +x

2. Obliczyc granice:

a)lim x!1 x − 1

2− p x ,

b)lim n!1 (1 + 1 n ) −2n

c)lim n!1 ( n 2 +2

n+3 ) 5−2n

f)lim n!1 ( n 2 +2

x−2 ,

b)lim x!3 x+2

q)lim x!3 x 2 −9

p x−1 ,

b)lim x!4 x−4

1− p −x ,

d)lim x!0 x+ p x

p x ,

e)lim x!0 x− p x

p x 2 +1−1

p x +1−1 ,

g)lim x!−2

p x 2 +21−5

x+2 ,

p x 2 +1− p x+1

h)lim x!0

1− p x+1 ,

p x 2 +1− 1

i)lim x!0

p x 2 +25−5 ,

p x 2 +x+1−1

j)lim x!0

3. Obliczyc granice:

a)lim x!0 1

(x−1) 2 ,

c)lim x!0 −2

x 4 ,

d)lim x!0 −3

|x|x 2 ,

e)lim x!2 −3

(x−2) 2 ,

f)lim x!1 7

|x−1| ,

g)lim x!−1 5

|x+1|(x+1) 3 ,

h)lim x!−1 18

(x+1) 2 ),

i)lim x!3 5

p |x−3| ,

j)lim x!0 −1

|x|x 4 ,

k)lim x!1 6

|x 2 −1| ,

l)lim x!−1 6

|x 2 −1 ,

m)lim x!2 −3

|−x 2 +4| ,

n)lim x!4 9

(x 2 −9)(x 2 −16) 2

4. Obliczyc granice:

a)lim x!1 (x 3 + 3x 2 −5x−1),

b)lim x!−1 (−x 5 −2x 3 + 4x 2 −5x−9),

c)lim x!1 (x 4 −5x 3 + 7),

d)lim x!−1 (x 2 + 3x−8)(4−x),

e)lim x!1 (−x 5 −x−4),

f)lim x!−1 (−x 7 + 2x−1),

g)lim x!1 (x−1)(x−2)(x+ 3)(x+ 4),

h)lim x!1 3x+1

2x 2 +3 ,

c)lim x!−1 −x−1

x + p x ,

f)lim x!0

|x| ,

b)lim x!0 1

5x+2 ,

i)lim x!1 7x 2 −1

j)lim x!1 ( 3

x+1 − 2x

4x+5 ),

x+2 − 2 x ),

l)lim x!1 3x 2 +5

4x 3 +3 ,

m)lim x!1 2x 4 +3

4x 2 +7 ,

n)lim x!1 4x 2 −5

2x 6 −3 ,

o)lim x!1 (x−1)(5−2x)

4x 2 +1 ,

p)lim x!1 x 2 −1

x 2 −3x+2 ,

q)lim x!1 x−2

x 2 −2x+1 ,

r)lim x!1 x 2 +1

2−2x 2 ,

s)lim x!1 5x 2 −3x+11

2−x+2x 2 ,

t)lim x!1 x 6 +2x+4

x 7 +2 ,

x+1 ,

v)lim x!−1

p −x+1

4 p 1−x ,

w)lim x!1 x−1

p 2+x+x ,

5. Obliczyc granice:

a)lim x!1 (1 + 1 x ) x ,

b)lim x!1 (1 + 1 x ) 2x ,

c)lim x!1 (1 + 1 5x ) x ,

d)lim x!1 (1− 1 x ) 5x ,

e)lim x!1 (1− 1 2x ) x ,

f)lim x!1 ( 2x+3

3x+4 ) 2x 2 ,

h)lim x!1 ( x 2 +2

x 2 +3 ) 2x 2 ,

i)lim x!0 (1 +x) 1 x ,

j)lim x!0 (1−3x) 1 x ,

k)lim x!0 (1 + 2x) 1 x

6. Obliczyc granice w punkciex 0 :

( 5x

x+1 dlax2<\{−1}

2 dlax 0 =−1

f(x) =

( 4x 2 +1

f(x) =

3−2x dlax2<\{ 3 2 }

1 dlax 0 = 3 2

k)lim x!1 ( 7x

p x

u)lim x!1

2x+5 ) 2x ,

g)lim x!1 ( 3x−1

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