Φυλλο Εργασιας - Κανονας de l hospital
DESCRIPTION
Φυλλο Εργασιας - Κανονας De L HospitalTRANSCRIPT
-
2o
1
:
31
: Del- Hospital
: ::..
:..:
,.!!!!!! ! ;;;; ;; ...
...
.
1) f g A(1,0)
.
2)
;; .
.
, .. :
. .
, , :
.
( Geogebra), :
.
, ZOOM .
,
() . .
21
lnlim
1x
x
x
( ) lnf x x= ( ) 21g x x=
: 1 y x= : 2 2 y x= +
01x =
( ) lnf x x= ( ) 21g x x=
1y x= 2 2y x= +
01x =
2
ln
1
x
x
1
2 2
x
x
+0
1x =
2
ln 1 1 1,
1 2 2 2( 1) 2
x x x
x x x
= =
+
01x =
( )
( )
( )
( )
1
1
f x f
g x g
( )
( )
( )
( )11
lim1x
f x f
g x g
=
0
1x =
( ) lnf x x= ( ) 21g x x= 1y x=
2 2y x= +
2ln 1y x y x= =
1y x= 2 2y x= +
(1,0)A
lny x= 1y x=
21y x= 2 2y x= +
-
2o
2
:
(ggb)
: ( 0
0 ) (
) ( de l Hospital).
0)(lim0
=
xfxx
, 0)(lim0
=
xgxx
, (0
lim ( )x x
f x
= , 0
lim ( )x x
g x
= ) 0
{ , }x +
0
( )lim
( )x x
f xL
g x
=
( ), :
0 0
( ) ( )lim lim
( ) ( )x x x x
f x f xL
g x g x
= =
.
: f , g : ) 0x
0x ) '( ) 0g x .
1) Del Hospital : 0
0
.
2) Del Hospital x ,0
x x
3) Del Hospital :
0 00 ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ,0 ,1 + + + + ,
0
0
.
4) 0
a
, 0
Del Hospital .
5) 0
( )lim
( )x x
f x
g x
0
0
', 'f g
, 0
''( )lim
''( )x x
f x
g x ...
6) 0
( )lim
( )x x
f x
g x
..
0
( )lim
( )x x
f x
g x (
!!!! ) limx
x x
x x
+
+
.
7) 0
0 , 1+
0
( )lim ( )g xx x
f x
=0
( ) ln ( )lim
g x f x
x xe
.
-
2o
3
:
8) 0 ( ) 0
0
( )lim
1
( )( )
lim( )
( )lim
1
( )
ox x
x x
x x
f x
g xf x
g x
g x
f x
=
+
0
0 0 0
( ) ( )lim( ( ) ( )) lim ( ) 1 lim 1
( ) ( )
( )1
( ) ( )lim( ( ) ( )) lim lim 1
1 ( )( )
o ox x x x x x
x x x x x x
g x g xf x g x f x
f x f x
g xf x g x
f x g xf x
f x
=
= =
1) :
i) )1ln(
lim
0 + x
x
x
0
0
ii) 4
2
0
1lim
x
x
x
0
0
iii) x
xx
x 1
lim
0
0
0
Iv) 0
2lim
x x
x
e e x
x x
(.dlh)
v) 2
lim
x
x
e
x+
( 2000
lim
x
x
e
x+
)
,vi) 0
1lim
lnx x x+
( ..,.
) , v) 0
lnlimx
x
x+
(
.)
2) :
4 4 4
99
( 2)( 2 3)......( 99 100)lim
( 1)x
x x x x x x
x
+ + +
( - )
3) ) : ) lim ( ln )x
x
e x
+
) 1 2 2
lim ( 5 2)x
x
e x x
+
+ (
0,
0
) ) 0
limx
x
x
+
0
0
, ) 1
lim 1
x
x x+
+
( )1 .
4) ) : ) lim
x x
x xx
e e
e e
+
+
, ) limx
x x
x x
+
+
(.dlh ??
. dlh . !!!!!)
-
2o
4
:
5) f 0x a= ,a ,
2
5a :
2
2
( ) 5 10lim
( )x a
f x x ax
x a
+
(
).
6) : :f '(0) (0) 2015f f= = .
20
( ) 2 ( ) ( )lim '( )h
f x h f x f x hf x
h
+ + = , . (
).
7)
=