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1
De L Hospital
f,g
0 0x x x xlim f x 0 lim g x
0 0x x x xlim f x lim g x ,
0
x ,
0x x
f xlim
g x
0 0x x x x
f x f xlim lim
g x g x
, , .
0x x
f xlim
g x
0x x
f xlim
g x.
0 0
0
0
0
1.
100
10x 1
x 10x 9lim
x 1
xx
xlim
e
x 2
2x 0
e 1lim
x 0
lim
0100100 990
10 9x 1 DLH x 1 x 110
x 10x 9x 10x 9 100x 10 100 10lim lim lim 9
10x 1 10xx 1
x xx DLH x x
x
xx 1lim lim lim 0
e ee
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2
)
22 2
0x 2
x 2 x0
2x 0 DLH x 0 x 0 x 02
e x 1 2e x 1 2xe 2xlim lim lim lim
2 x xxx
2
xx e 1
2
2x
x 0
e 1lim 0
xx xx
x
0 0
0 0
DLH DLHx 0 x 0 x 0
x xx x x x x 2 x x xlim lim lim
x x 1 x xx x
x 0
1lim 2 x x x
x
x 0
lim 2 x x x 2
x 0
1lim
x
x 0
lim x 0 x 0
x 0,2
.
2.
x3 3
, x 1f x x 1
3ln3, x 1
.
f .
f .
0xx x0
x 1 x 1 DLH x 1 x 1
3 33 3 3 ln3limf x lim lim lim 3ln3 f 1
x 1 1x 1
f x 1.
f ,1 1,
f .
x0
x 0
2x 1 x 1 x 1 DLH
3 33ln3f x f 1 3 3 3ln3 x 3ln3x 1lim lim lim
x 1 x 1 x 1
0
x x 2 20
x 1 DLH x 1
3 ln3 3ln3 3 ln 3 3ln 3lim lim
2 x 1 2 2
23ln 3
f 12
.
H f x 1
x x
2
3 ln3 x 1 3 3f x
x 1f
x x
2
2
3 ln3 x 1 3 3, x 1
x 1f x
3ln 3,x 1
2
.
3. ,
x
x
e 1, x 0f x
e x, x 0
0
x 0 .
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3
f 0
x 0
x x
x 0 x 0 x 0 x 0
lim f x lim f x f 0 lim e 1 lim e x 1 1 1 0
x 0 x 0
f x f 0 1lim lim
x
1 0
x
0xx x0
DLHx 0 x 0 x 0 x 0
e x 1f x f 0 e x 1 elim lim lim lim 1
x x 1x
,
f 0
x 0
x 0 x 0
f x f 0 f x f 0lim lim
x x
1 0 1.
4. f :
xlim f x f x 2
xlim f x
xlim f x 2
xlim f x 0
)
xx x x
x xx x DLH x xx
e f xe f x e f x e f xlim f x lim lim lim
e ee
x
x x
elim f x lim
x
f x f x
e 2
f x f x g x f x g x f x
x xlim f x lim g x f x 2 2 0
0
5.
x
xlim xe
2
x 0
lim x lnx
0x
x xx x DLH x xx
xx 1lim xe lim lim lim 0
e ee
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4
202
DLHx 0 x 0 x 0 x 0 x 02
2 3
1lnxlnx xxlim x lnx lim lim lim lim 0
1 2 2x
x x
6.
x 2
xlim e x x
xlim x lnx
x
xlim ln e 1 x
x
x 2 2
2x x
e 1lim e x x lim x 1
xx
x x x
2x DLH x DLH x
e e elim lim lim
2x 2x,
x
2x
e 1lim 1
xx
2
xlim x
x x
lnxlim x lnx lim x 1
x
x DLH x
1
lnx xlim lim 0x 1
)
x
x x x
xx x x
e 1lim ln e 1 x lim ln e 1 lne lim ln 0
e
xx x
xx DLH x xx
e 1e 1 elim lim lim
ee
xe
1
0 0
0 , 0 , , 1
7.
x
x 0
lim x
1
x
xlim x
1
x
xlim lnx
x 0
lim x
00
x xlnx
x 0 x 0
lim x lim e .
0
DLHx 0 x 0
lnxlim xlnx lim
1
x
x 0 x 0
2
1
xlim lim x 01
x
,
xlnx u x u 0
x xlnx u 0
u 0x 0 x 0
lim x lim e lime e 1.
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5
x 0
1
x
1 1 lnxlnx
lnxx x xx e e e lnx
ux
.
x x DLH x
1
lnx xlim u lim lim 0x 1
1 lnx
u 0x x
x x u 0lim x lim e lime e 1
x 0
1
xln lnx1ln lnx1 ln lnx
x xxlnx e e e
ln lnx
ux
.
x x DLH x
1 1ln lnx lnx xlim u lim lim 0
x 1
ln lnx1u 0xx
x x u 0lim lnx lim e lime e 1
x 0,2
xx ln x xln xx e e xln x u .
DLHx 0 x 0 x 0 x 0 x 0
2
x
ln x x xxlim u lim xln x lim lim lim 0
1 1 x
x xx
x xln x u
u 0x 0 x 0
lim x lim e lime 1
8. g .
h 0
g x g x hg x lim
h.
g
3
2h 0
g x h 2g x g x hlim 20x 6x g 0 g 0 1
h
i. 5 3g x x x x 1.
ii. g -1.
) g
h 0
g x h g xg x lim
h.
h u
h 0 u 0 u 0 u 0
g x g x h g x g x u g x u g xlim lim lim g x
h u u.
i.
0
0
2h 0 h 0
g x h 2g x g x h g x h g x hlim lim
2hh
h 0
g x h g x g x g x h1lim
2 h
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6
h 0
g x h g x g x g x h1lim
2 h h
1
g x g x g x2
.
3
2h 0
g x h 2g x g x hlim 20x 6x
h
3g x 20x 6x 4 2 4 2
g x 5x 3x g x 5x 3x c , c .
g 0 1 c 1 4 2g x 5x 3x 1, x .
5 3g x x x x 5 3
1g x x x x c ,
1c .
1
g 0 1 c 1 5 3g x x x x 1, x .
ii. 4 2g x 5x 3x 1 0 g -1.
9. 2f x x 2lnx, x 0 .
) f x 1 x 0 .
) f.
)
lnx, x 0
f xg x
k, x 0
.
i. k g
ii. 1
k2
g
0,e .
) f 0,
22 x 12
f x 2xx x
.
2
22 x 1
f x 0 0 x 1x
x 1.
x 0,1 f x 0 f
0,1 .
x 1, f x 0 f 1, .
f 2f 1 1 2ln1 1 f x 1 x 0 .
)
2
x o x o
lim f x lim x 2lnx
x o
lim lnx x 0 ,
y y f
C .
x x
f x lnxlim lim x 2
x x
x DLH x x
1
lnx 1xlim lim lim 0x 1 x
fC
x 0 1
f - +
f ] [
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2 2
2x x x
lnxlim f x lim x 2lnx lim x 1 2
x.
2 2x DLH x x
1
lnx 1xlim lim lim 02xx 2x
xlim f x
fC
) i. g 0,
g
x o
lim g x g 0 k
2 2 2DLHx o x o x o x o x o
1 1
lnx 1 1x xlim g x lim lim lim lim2 2x 2lnx 2x 2 2x 2
2xx x
1
k2
.
ii.
2 2
lne 1 1g e 0
f e e 2lne e 2
1g 0 k 0
2
g 0 g e 0 g 0,e
Bolzano g x 0 0,e .
10. 2
f x ln 1 x x 1 ln x 2 , x 1
1.
xlim f x
1
i. f
ii. f.
iii. 2f x 0 0 .
)
2
21 x x 1
f x ln 1 x x 1 ln x 2 lnx 2
21 x x 1
ux 2
.
1 0 1
2 2
x x x x
1 x x 1 1 xlim u lim lim lim 1 x
x 2 x
x xlim f x lim lnu .
1 0 1
x x x
x 1 xlim u lim lim 1
x 2 x
x xlim f x lim lnu 0 .
xlim f x 1.
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8
1 f x ln x 1 ln x 2 , x 1.
i. f
1 1 1f x
x 1 x 2 x 1 x 2.
f x 0 x 1 f
x 1 x 1
lim f x lim ln x 1 ln x 2
x 1
lim ln x 1
x 1
lim ln x 2 0
xlim f x 0 f
x 1 x
f A f 1, lim f x , lim f x ,0 .
ii.
x 1
lim f x x 1 f
C .
xlim f x 0 y 0
fC .
iii. 20 0 2
f.
f 2 2f x f x 0
0 .
11. f : f 0 f 0 0 ,
xe f x f x 1 f x xf x x .
xf x ln e x , x .
xln e x x
0,
2.
) x x x xe f x f x 1 f x xf x e f x e f x e xf x
x x x x x xe f x e xf x e f x e xf x c e f x xf x e c, c
(1)
x 0 0 1 c c 1
x x x xe f x xf x e 1 e x f x e 1 (2)
xg x e x, x g x
g x e 1.
x xg x 0 e 1 0 e 1 x 0 .
x 0 g x 0 g 0,1 x 0
g x 0 g ,02 g g 0 1
g x g 0 1 x g x 0
xxx e x 1 0x
x x
e xe 1f x f x ln e x c , c
e x e x
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9
x 0 f 0 c c 0 xf x ln e x , x .
xx e x 0x x
x
e 1f x 0 0 e 1 0 e 1 x 0
e x
x 0 f x 0 f 0,1 x 0
f x 0 f ,02 f f 0 0
f x f 0 0 x ,
x xx x
x x x 2x
e x x 1 e 1e 1 e x x 1 x 1f x 1
e x e x e x e x
x x x x x
2 2x x
e x xe x e 1 2e xe 1f x
e x e x
x xh x 2e xe 1, x h
x x x xh x 2e e xe e 1 x .
xh x 0 e 1 x 0 1 x 0 x 1
x 1 h x 0 h ,01 x 1
h x 0 h 0,2 .
x
x xx x DLH x
x 1lim xe lim lim 0
e e
x x
x xlim h x lim 2e xe 1 1
x x x
x x xlim h x lim 2e xe 1 lim e 2 x 1 h 1 e 1
1,0 h
1x
h lim h x ,h 0 1,e 1 .
0 1,e 1 1
x ,0 1h x 0 h ,01
1
x h ,0
20, h
2x
h h 0 , lim h x 1, .
0 1, 2
x 0, 2h x 0 h 0,2
2
x h 0, .
h
1 1x x h x h x 0 f x 0 f
1
1,x .
h
1 1x x 0 h x h x h x 0 f x 0 f
1
1x ,0 .
f 1 1x ,f x .
h
2 20 x x h x h x 0 f x 0 f
2
20,x .
h
2 2x x h x h x 0 f x 0 f
2
2x , .
f 2 2x ,f x .
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10
xx ln e x x, x 0,
2.
0 1 0
2ln e2 2
g 0,1
2 20 g g 0 e 1 ln e ln1 02 2 2 2 2
0 02
0,
2Bolzano
3x 0,
2
3x 0 .
0,
2
x
x
e 1x x 0
e x
x
x
e 1f x 0
e x
x 0
x 0,2
0,
23
x
x 0 .
12. f : xxf x 1 e
x .
xe 1
, x 0f x x
1, x 0
f
A 0,f 0
2f x x 2 , x
x 0
lim x lnx ln f x .
x xxf x e xf x e 1 x 0
xe 1
f xx
.
f 0
x 0
0xx x0
x 0 x 0 DLH x 0 x 0
e 1e 1 ef 0 lim f x lim lim lim 1
x 1x
xe 1
, x 0f x x
1, x 0
f x 0
x x x x
2 2
e x e 1 e x e 1f x
x x.
x xg x e x e 1, x .
g x x x xg x e x e e xe .
x 0 g x 0 g 0, .
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11
x 0 g x g 0 0 f x 0 f
0, .
x 0 g x 0 g ,0 .
x 0 g x g 0 0 f x 0 f
,0 .
f -
f f.
xx x
x x DLH x x
e 1e 1 elim f x lim lim lim
x 1x
x
x x
e 1lim f x lim 0
x
x
xlim e 1 0
x
1lim 0
x
1f x x
A f A lim f x , lim f x 0,
x0
xx 0
2x 0 x 0 x 0 DLH x 02
e 11 e 1 xf x f 0 e 1 xxlim lim lim lim
x 0 x xx
x x
x 0 x 0
e 1 1 e 1 1lim lim
2x 2 x 2
1
f 02
fC
1 1
y f 0 f 0 x 0 y 1 x y x 12 2
.
f
1
f x x 1 2f x x 22
f x x x 0 .
DLHx 0 x 0 x 0 x 0 x 0
2
1lnxlnx xlim x lnx lim lim lim lim x 0
1 11
x xx
f x u
u 1 u 1x 0
lim ln f x limlnu 0 (
x 0lim f x 1
x 0 x 0
lnxlim x lnx ln f x lim ln f x 0
1
x
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12
13.
x x
x 0
e e 2xlim
x
3
xx
x lnxlim
e
1
x
x 0
lim xe
14. f 3 2 2x x f x
f 0 .
15.
x 0
lim xlnx
x 0
lim
1
x
xlim x e 1
16.
2
xlim 3x 2lnx
x 0
1 xlim
x 1
17.
x 1
x 1
lim x 1
2x
x
x 4lim
x 5
1
x
xlim x 1
18. f
2h 0
f x 2h f x 2h 2f xlim f x
4h
2h 0
f x 2h 2f x f x 2hlim 4f x
h
19.
-x 1
x x 1f x
1- e ln x 1 x 1,2.
x 1
x 1
1 elim
x 1.
f 0
x 1.
1 1,2
f A ,f
x x.
20.
2x , x 0
f x x , 0 x 1
1 xlnx, x 1
, .
f
1 0
i.
2x
f xlim
x
ii. Να υπολογίσετε τα όρια:
x 1
f x f 1lim
x 1 ,
x 1
f x f 1lim
x 1 .
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13
21. xlnx x 1 x 0 .
2 22x lnx 3x 4x 1 0
22. 2ln x 2xlnx 1 0, .
23. f :
22xf x x f x 3 f x x
1f 1
2
3
2
xf x , x
x 1f
.
f
3 2
2 2f 5 x 1 8 f 8 x 1
xlim f x 5 .
24.
2x x 2
h x , x 1,x 1
y x 2 h
1.
i. y x 2
h .
ii. h.
4x 3
h x 0x
1,0 .
25. f : 0,
f x xf x 2x x 0, f 1 10 .
2
x 9f x , x 0
x
f.
f .
f x 10 x 1 f x x 1.
