x0R x0 R

1

http://www.perikentro.blogspot.gr/ : - 1 -

1.1. f:[0, ) IR f(x)=x2-4. . f 1-1 . f . f f-1 . , : ). )x(flim

2x ). )x(flim 1

4x

). )x(flim 1

0x

. x0 .. f f-1

: )x(flim)x(flim 1xxxx 00

1.2. f 1)h3(flim

0h

:

. )x(flim3x

. : 3)2)x(f(lim3x

1.3. f(x)=2xx3x2

. x

)x(flim

0x

1.4. :

). 3x4x

1xxxlim

2

23

1x

). 6x

33xlim

6x

). 2x

4x4xlim

4x

v).

1|x|1|2x|

lim1x

1.5. 2-|x-1| f(x)-2xx2-2x+3 IRx )x(flim

1x

1.6. IRIR:f f2(x)+2f(x)+2x0 IRx , )x(flim

0x

:

IRx0 1

http://www.perikentro.blogspot.gr/ : - 2 -

2.1. :

). 1x

1xlim

31x

). 1|1x||2x|x

lim2x

2.2. 21x

xxlim

2

2

1x

, .

2.3. ( )

0xlim

f(x)

0x

21

x3

0x1x

11x

)x(f

2.4. IRIR:f :

32x

5x)x(flim

2x

2x6)x(xf

lim2x

2.5. f,g:(1,2) IR)3,2(

62x3x

)x(flim

22x

1)]4x)(x(g[lim 2

2x

.

)]x(g)x(f[lim2x

2.6. 1x2)x(fx2x 22 x IR :

1x

3)x(flim

1x

2.7. . 0|)x(f|lim

0xx

f x0 0)x(flim

0xx

. 0)x(flim 2

xx 0

f x0 0)x(flim

0xx

. IRIR:g,f 0)]x(g)x(f[lim 22

xx 0

)x(flim0xx

0)x(glim0xx

: IRx0 2

http://www.perikentro.blogspot.gr/ : - 3 -

3.1. :

)x3

x21x

xx(lim 2

20x

3.2. :

). x

xlim

3

0x

). xx

lim2

0x

).xx3xx2

lim0x

v). 1x

)1x(lim

21x

v). x

xx3lim

0x

v). xx

x21lim

20x

3.3. f:IR IR : |x2f(x)-2x| 3x4 IRx . ).x(flim

0x

3.4.

0x0

0xx1

x)x(f2

( ) : ). )x(flim0x

). x

)x(flim

0x

3.5. |x|

http://www.perikentro.blogspot.gr/ : - 4 -

4.1. :

). 1x

1xlim

2

2

1x

). 25x

1x3lim

25x

). 25x10x

x25lim

2

2

5x

). |x||x|

2x3lim

0x

4.2. 23

2

x xx

1xlim

4.3. 24xx2)x(f

lim0x

=- ).x(flim0x

4.4. 0)x(flim1x

21x )1x(

)x(flim 0

4.5. : |xx2x|

x|1x2|lim

231x

4.6. f g :

)x(f5x

lim3x

2x

)x(glim

3x. ( )

)]x(g)x(f[lim3x

4.7. z=+i, w=-i , IR

1x)(x

3x)x(f

2

. ,

z+w :

)x(flim1x

4.8. IRIR:f xy)y(xf)x(yf)y(f)x(f

IRy,x . 30x x

)x(flim

: IRx0 4

http://www.perikentro.blogspot.gr/ : - 5 -

5.1. :

. 1x2

2x3x4lim

2

x

.

|1x|x

3|xx|lim

2

2

x

.|4x|

2|x|x3lim

2

2

x

5.2. 2x

1x4x3)x(f

2

x

)x(flim

x , IR

).x)x(f(limx

5.3. 2x

)x(flim

x

:

1x5)x(f)x(f2x3

limx

5.4. f: IR)0,( :

2x

)x(flim

x

3]x2)x(f[lim

x

*IR :

1x2)x(fx

1x)x(f2lim

2x

=1

5.5. :

i. 1x

xxlim

2x

ii. x2)x1x(lim 2x

iii. )xx2(limx

iv.

1

x1

xlimx

v. x52x

xlim

2x

vi.

3x2x1

xlim

2

x

5.6 IR),0(:g,f 0)]x(g2)x(f3[lim 22

x

)x(flimx

0)x(glimx

5.7. IRIR:g,f : 0)]x(g)x(f[lim

x

0)]x(g)x(f[limx

. :

)x(flimx

0)x(glimx

: 5

http://www.perikentro.blogspot.gr/ : - 6 -

6.1. :

. 1354

234lim

x2x

1xx

x

.

1xx

x1x

x 2e

26e4lim

6.2. :

2005x8

2006x3lim

3x

6.3. :

. ).1xxln(lim 2x

. 1x1x

x

2

elim

. )x1xln(lim 2x

. x21x

lnlim2

x

6.4. 2x1

1)x(f

. g(x)=xf(x)2x

).x(glimx

6.5. i. :

=1x

xxlim

2x

= )x2

|x(|limx

ii. z=+i , i. w=+i, , IR . w z 2 : 2+2+4=0

6.6. IR),0(:f 3x

)x(flim

x

4)x3)x(f(limx

. IR 21xx3)x(xf

2x)x(flim

2x

: 6

http://www.perikentro.blogspot.gr/ : - 7 -

7.1 IRIR:f 1x

1xx2x)1()x(f

2

24

i. =1 C : )x(flimi3610)x(flimzz4

0xx z

ii. 1 w

i)x(flim2x

)x(flimw

1x2x

iii. ,|z|x

1|z|xlim

|z|x

3i1z

7.2. z

: IR2x

8x2x|i43z|lim

23

2x

7.3. z ,w : |z+2-2i|= |w-1+i|=

,>0. 74x

6xxlim

2

2

2x

:

i. ii. z w iii. z ,w |z-w|

7.4. IRx,Cz,|z||zx|

|z||zx|)x(f *

22

22

i. 20x |z|

)zRe(x)x(f

lim

ii. z Cf (2,1) 7.5. z=+i , IR 0z

2x3x

ziz2x5xizz)x(f

2

2

. 3)x(flim

1x

i. izz2ziz

ii. >0

: 7

http://www.perikentro.blogspot.gr/ : - 8 -

1. f(x)> , IR x0

)x(flim

0xx

2.

|)x(f|lim0xx

, IR

)x(flim0xx

)x(flim0xx

3.

0|)x(f|lim0xx

0)x(flim0xx

4. IR)]x(g)x(f[lim0xx

)x(flim0xx

)x(glim0xx

5. 0)x(flim0xx

0)x(f x0

6. f x0 )x(flim

0xx

7. IR)x(g)x(f

lim0xx

0)x(glim0xx

0)x(flim0xx

8.

)x(flim0xx

)x(glim0xx

0)]x(g)x(f[lim0xx

9. 0)x(flim0xx

)x(f

1lim

0xx ( 0)x(f x0)

10.

)x(flim0xx

f x0

11. P(x) , , IR P()P()