Όρια Προτεινόμενες Ασκήσεις
TRANSCRIPT
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x0R x0 R
1
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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
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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
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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|
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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
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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
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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
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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
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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()