Κ16-Κανόνες Hospital (6)
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Transcript of Κ16-Κανόνες Hospital (6)
-
7/25/2019 16- Hospital (6)
1/13
!"#$%"&'($. )*+,-.*/0*&
!"#$%$&'()"#$"&*+,-
.&/0*1201(34*5&1&67681/."-69*5&$81/:655"71-8+$(
!"#$%"&'()
*+,-(./0-
!"#$%&'#()*- +%,()%-2015 - 2."/'%0.
!"#$#%&'()de l' Hospital
10!"!#$%&'(#$)"*+!,'-,*&+
!"#$%&'(" 16
!"#$%&' )*%+*"$' *,) -.$) /"
0.+*1 ,2# %)#-#2# ,34 del'Hospital.
-
7/25/2019 16- Hospital (6)
2/13
!"#$%&'()*+,-.,%("/#$/$1).
12 + 1 01'('/(,2!3'4,53#*6,73/*, 894".'3,2,- :;6-#9$'(3#":' 6-#9
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7/25/2019 16- Hospital (6)
3/13
1. f(x)=
!"#x!1
x2, $# x< 0
%x+1
2ex, $# x" 0
#
$
%%%%%
&
%%%%%
. ,
.
2. f(x)=
x! !nx
1"x
, !" 0< x# 1
"1 , !" x= 1
$
%
&&&&
'
&&&&
, :
) f . ) !f(1)="1
2.
3. !,"! ! , f(x)=1+! !"#$x,!$ x"
%
3
&+ 'x
2
, !$ x>%
3
#
$
%%%%%
&
%%%%
%
!
3 .
4. f (0,+!) , ! imx!+"
[f(x)+ x# $f(x)]= 1993 .
! imx!+"
f(x) # "* , ! imx!+"
f(x)= 1993 .
5. f:!! ! , f5(x)+ 5f(x)= x! !x, -
x! ! , ! imx!0
f(x)
x
3.
6. f:!! ! ,
x!f(x)+e!x=f(x) !!x+ex, x! ! .
7. !,",#! ! , ! imx!0
! "ex
+ ""e#x
+ #x
x# $x= 4 .
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8. f 0 f(0)= 0 , !f(0)= 1 ,
! imx!0
x"f(x)
!x#xex.
9. f:!! !
f5(x)+f3(x)+f(x)= x! !x, x! ! , ! imx!0
f(x)
x3=
1
6.
10. f:!! ! ,
xex![f(x)+ 2]= ex ![f(x) !!x+ex]"1 , x! !* .
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7/25/2019 16- Hospital (6)
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1. f(x)=
!"#x!1
x2, $# x< 0
%x+1
2ex, $# x" 0
#
$
%%%%%
&
%%%%%
. ,
.
x< 0 , f -
,
!f(x)=(!"#x"1 !) #x2 "(!"#x"1) #(x2 !)
(x2)2=
"$x#x2 "(!"#x"1) #2x
x4 $
! "f(x)=
#x$ [x$ !x+ 2("#$x#1)]
x4 ! "f(x)= #
x$ !x+ 2"#$x# 2
x3.
x> 0 , f -
, !f(x)= !"#x+1
2ex .
0 , '
, ! imx!0
f(x)=f(0)= !0+1
2e0 =
1
2.
! imx!0"
f(x)=! imx!0"
!"#x"1
x2=
0/0
! imx!0"
(!"#x"1 #)
(x2 #)=! im
x!0"
"$x
2x="
1
2! imx!0"
$x
x="
1
2$1="
1
2.
! imx!0+
f(x)= ! imx!0+
!x+1
2ex
"
#$$$$
%
&
''''= !0+
1
2e0 =
1
2.
! imx!
0
f(x) , f 0,
0.
!f(x)="
x# !x+ 2"#$x" 2
x3, %$ x< 0
"#$x+1
2ex, %$ x> 0
$
%
&&&&&
'
&&&&&
.
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
789:;:'3.!"#$#%&'()de l' Hospital.
)23"2$51-+$
-
7/25/2019 16- Hospital (6)
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2. f(x)=
x! !nx
1"x, !" 0< x# 1
"1 , !" x= 1
$
%
&&&&
'
&&&&
, :
) f . ) !f(1)="1
2 .
!) x> 0 , x! 1 , f
.
! imx!1
f(x)= ! imx!1
x " !nx
1#x=
0/0
! imx!1
(x" !nx $)
(1#x $)= ! im
x!1
(x) " !nx+ x" (!nx $)
#1=
=!! imx"1
!nx+ x#1
x
$
%&&&&
'
(
))))=!! im
x"1(!nx+1)=!(!n1+1)=!1=f(1)* ! im
x"1f(x)=f(1) .
f 1, .
") !f(1)= ! imx"1
f(x)#f(1)
x#1= ! im
x"1
x$ !nx
1#x+1
x#1= ! im
x"1
x$ !nx+1#x
#(x#1)
x#1=
=!! imx"1
x# !nx+1!x
(x!1)2=
0/0
! ! imx"1
(x # !nx+1!x $)
[(x!1)2 $]=!! im
x"1
(x $) # !nx+ x# (!nx $) !1
2(x!1) #(x!1 $)=
=!! im
x"1
!nx+ x#1
x
!1
2(x!1)=!! im
x"1!nx+1!12(x!1)
=!12! imx"1
!nx
x!1=
0/0
!12! imx"1
(!nx $)(x!1 $)
=
=!
1
2! imx"1
1
x
1=!
1
2! imx"1
1
x=!
1
2#1
1$ %f(1)=!
1
2.
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789:;:'3.!"#$#%&'()de l' Hospital.
)23"2$51-+$
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3. !,"! ! , f(x)=1+! !"#$x,!$ x"
%
3
&+ 'x
2, !$ x>
%
3
#
$
%%%%%
&
%%%%%
!
3.
f !
3,
!
3.
! imx!
!
3
"
f(x)= ! imx!
!
3
"
(1+" ##$%x)= 1+" ##$%!
3= 1+" #
1
2= 1+
"
2.
! imx!
!
3
+
f(x)= ! imx!
!
3
+
"+ #x
2
"
#$$$$
%
&
''''= "+ #
!
32= "+ #
!
6= "+
1
2.
f
!
3
!
"####
$
%
&&&&= 1+" '#$%
!
3 = 1+" '
1
2 = 1+
"
2 .
1+!
2= "+
1
2! 2+! = 2"+1! ! = 2""1 (1)
!1= ! im
x!!
3
"
f(x)"f !
3
#
$%%%%
&
'
((((
x" !
3
= ! imx!
!
3
"
1+" )#$%x"1" "
2
x" !
3
= ! imx!
!
3
"
" #$%x"1
2
#
$%%%%
&
'
((((
x" !
3
=
=! !! imx"
"
3
#
#$%x#12
x# "
3
=
0/0
! !! imx"
"
3
#
#$%x#12
$
%&&&&
'
())))
*
x# "
3
$
%&&&&
'
(
))))
*=! !! im
x""
3
#
#&x
1=#! !&
"
3+
! !1="
! 3
2.
!2= ! im
x!!
3
+
f(x)"f !
3
#
$
%%
%%
&
'
((((
x"!
3
= ! imx!
!
3
+
"+ #x
2 "1"
$
2
x"!
3
=(1)
0/0
! imx!
!
3
+
"+ #x
2
"1"$
2
#
$
%%
%%
&
'
((((
)
x"!
3
#
$%%%%
&
'
((((
)*
! !2= ! im
x"!
3
+
"#$
x
2#x
2
$
%&&&&
'
(
))))
*
1=
1
2! im
x"!
3
+
"#$
x
2
$
%&&&&
'
(
))))=
1
2#"#$
!
3
2=
1
2#"#$
!
6=
1
2# 3
2!
! !2=
3
4
.
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
789:;:'3.!"#$#%&'()de l' Hospital.
)23"2$51-+$
-
7/25/2019 16- Hospital (6)
8/13
f !
3, !
1=!
2!"
! 3
2=
3
4! !="
1
2.
(1) !1
2= 2!!1" 2!=
1
2" !=
1
4.
4. f (0,+!) , ! imx!+"
[f(x)+ x# $f(x)]= 1993 .
! imx!+"
f(x) # "* , ! imx!+"
f(x)= 1993 .
! imx!+"
[f(x)+ x# $f(x)]= 1993% ! imx!+"
[(x $) #f(x)+ x# $f(x)]= 1993%
! ! im
x"+#[x$f(x) %] = 1993 .
! imx!+"
f(x)= ! imx!+"
x#f(x)x
.
! imx!+"
f(x) # "* , +!
+!,
! imx!+"
f(x)= ! imx!+"
[x#f(x) $]
(x $)= ! im
x!+"
[x#f(x) $]
1= ! im
x!+"[x#f(x) $] % ! im
x!+"f(x)= 1993 .
5. f:!! ! , f5(x)+ 5f(x)= x! !x, -
x! ! , ! imx!0
f(x)x3
.
!"#$%&'()'*#%+,"-./01&20&0!!
!"#$#%"#&"'()'*#$(+,-(+./$+01"+/2#$3,/"#&1+'%"+1"'(3#(4#"&+"+5/3%+56), &17&/"$#5&+8+93'*(-$(":f(0). !"' #5(0, ;#
7")0".:28&.@2:- www.mathsteki.gr
- 7 -
-
7/25/2019 16- Hospital (6)
9/13
f5(x) ! f(x) x5 ,
! .
, !
, -
5f4(x) ! "f(x)+ 5 "f(x)= 1#!"#x$ "f(x) ![5f4(x)+ 5]= 1#!"#x.
5f4(x)+ 5> 0 , f(x),
!f(x)=1"!"#x
5f4(x)+ 5.
, ! = ! imx!0
"f(x)
3x2= ! im
x!0
1#!"#x
5f4(x)+ 5
3x2
= ! imx!0
1#!"#x
3x2
[5f4
(x)+ 5]
= ! imx!0
1#!"#x
15x2
$[f4
(x)+1]
%
! ! =1
15! imx"0
1#!"#x
x2 $[f4(x)+1]! ! =
1
15! imx"0
1#!"#x
x2 $
1
f4(x)+1
%
&
''
(
)
**.
! imx!0
1"!"#x
x2
=
0/0
! imx!0
(1"!"#x #)
(x2 #)=! im
x!0
$x
2x=
1
2! imx!0
$x
x
=
1
2$1=
1
2.
! =1
15!1
2!
1
f4(0)+1=
(2) 1
30!
1
04 +1" ! =
1
30.
6. f:!! ! ,
x!f(x)+e!x=f(x) !!x+ex, x! ! .
x!f(x)"f(x) !!x=ex"e!x#f(x) !(x" !x)= ex"e!x .
x! 0 x!!x" 0 , f(x)=ex!e!x
x! !x,
!*
, f ! , 0,
! imx!0
f(x)=f(0) .
! =! imx!0
f(x)=! imx!0
ex"e!x
x" !x=
0/0
! imx!0
(ex"e!x #)
(x" !x #)=! im
x!0
ex"e!x$(!x #)
1""#$x%
! ! =! imx"0
ex
#e!x
$"#$x
1#"#$x=
0/0
! imx"0
(ex#e!x $"#$x %)
(1#"#$x %)=! im
x"0
ex
#(e!x %) $"#$x#e!x $("#$x %)
!x!
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
789:;:'3.!"#$#%&'()de l' Hospital.
)23"2$51-+$
-
7/25/2019 16- Hospital (6)
10/13
! ! = ! im
x"0
ex
#e!x$"#$x$"#$x+e
!x$!x
!x= ! im
x"0
ex
#e!x$"#$
2x+e
!x$!x
!x!
0/0
! ! = ! imx"0
(ex#e!x $"#$2x+e!x $!x %)
(!x %).
(ex!e!x ""#$2x+e!x "!x #) =
= ex
!(e!x ") #"#$2x!e!x #("#$2x ") + (e!x ") #!x+e!x #(!x ") =
= ex
!e!x
""#$x ""#$2x!e
!x"2"#$x "("#$x #) +e!x ""#$x "!x+e!x ""#$x=
= e
x
!e!x
""#$3x+e
!x"2"#$x" !x+e
!x"!x""#$x+e
!x""#$x=
= ex
!e!x
""#$3x+ 3e!x " !x""#$x+e!x ""#$x= ex!e!x "(3!x+1!"#$2x)=
= ex
!
e
!x
"
"#$x"(3!x+ !
2x)= e
x
!
e
!x
"
!x"
"#$x"(3+ !x) .
! = ! imx!0
ex
"e!x#!x#"#$x#(3+ !x)
"#$x=
e0"e
!0#!0 #"#$0 #(3+ !0)
"#$0$
! ! =! im
x"0f(x)= 1=f(0) .
f(x)=
ex!e!x
x! !x, "# x" 0
1 , "# x= 0
#
$
%%%%
&
%%%%
.
7. !,",#! ! , ! imx!0
! "ex
+ ""e#x
+ #x
x# $x= 4 .
f(x)=! !ex+ "!e"x+ #x
x" $x, ! im
x!0f(x)= 4 .
! !ex+ "!e"x+ #x= (x" $x) !f(x) ,
! imx!0
(! "ex+ ""e#x+ #x)= ! imx!0
[(x# $x) "f(x)]$ ! "e0 + ""e0 + #"0 = (0#$0)"4$
! !+ "= 0! "="! (1)
(1), f(x)=!ex!!e!x+ "x
x! #x,
! = ! im
x!0f(x)= ! im
x!0
!ex"!e"x+ "x
x" #x=
0/0
! imx!0
(!ex"!e"x+ "x #)
(x" #x #)= ! im
x!0
!ex"!e"x $("x #) + "
1"$%&x%
! ! = ! imx"0
!ex
+!e#x
+ "
1##$%x.
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
789:;:'3.!"#$#%&'()de l' Hospital.
)23"2$51-+$
-
7/25/2019 16- Hospital (6)
11/13
g(x)=!e
x+!e
!x+ "
1!#$%x, ! im
x!0g(x)=! = 4 .
!ex+!e!x+ "= (1!#$%x) "g(x) ,
! imx!0 (!e
x
+!e"x
+ ")= ! imx!0 [(1"#$%x)#g(x)]$ !e0
+!e0
+ "= (1"#$%0)#4$ 2!+ "= 0$
! !="2" (2)
(2), g(x)=!e
x+!e
!x!2!
1!"#$x=
!(ex+e!x!2)
1!"#$x,
! = ! im
x!0g(x)= ! im
x!0
!(ex+e"x"2)
1""#$x= ! # ! im
x!0
ex+e
"x"2
1""#$x=
0/0
! # ! imx!0
(ex+e"x"2 $)
(1""#$x $)%
! ! = ! " ! imx
#0
ex
+e$x"($x %)
"x
= ! " ! imx
#0
ex
$e$x
"x
=
0/0
! " ! imx
#0
(ex$e$x %)
("x %)
= ! " ! imx
#0
ex
+e$x
#$%x
!
!! = ! "e
0+e
0
"#$0! ! = 2! .
! = 4 , 2!= 4! != 2 .
!= 2 , (2) !=!4 (1) !=!2 .
8. f 0 f(0)= 0 , !f(0)= 1 ,
! imx!0
x"f(x)
!x#xex.
!"#$#%&!! !"#$%&'#()$*+(#'*,&%-,%,(-+%(&.$"/0&%$*+(12%(+%1,/$324"2%5324", 2&"*(+6"$&.$+-17,("#$"-"$,(8de l' Hospital, 2%-,%2&"2/"&,$%-,%3f &/"$%*$+$979/1%31&2%51,3$, $))51&13&/(!!
!f(0)= 1 ! imx!0
f(x)"f(0)
x"0= 1# ! im
x!0
f(x)
x= 1 (1)
! =!
imx!0
x"f(x)
x2
!x#xex
x2
$ !=!imx!0
f(x)
x
!x#xex
x2(2)
! imx!0
!x"xex
x2
=
0/0
! imx!0
(!x" xex #)
(x2 #)=! im
x!0
"#$x"(x #) $ex"x$ (ex #)
2x=
=! im
x!0
!"#x"ex
"xex
2x=
0/0
! imx!0
(!"#x"ex"xex #)
(2x #)=! im
x!0
"$x"ex
"ex
"xex
2=
=
1
2
! imx!0
("!x"2ex"xex)=1
2
("!0"2e0"0 #e0)="1 .
'()*+(,-.*. /012340560,, 789(:(*),/89(8*;7#?&?-3.:$"-"&;,(8de l' Hospital.
/89(8*;731*@7;- www.mathsteki.gr
- 10 -
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7/25/2019 16- Hospital (6)
12/13
(1), (2) ! =1
!1" ! =!1 .
9. f:!! !
f5(x)+f3(x)+f(x)= x! !x, x! ! , ! imx!0
f(x)x3
=16
.
!"#$#%"#&"'()'*#$(+,-"+./#$01."#&-+'%"+-"'(0#(2#"&+"+3.0%+45), &-6&."$#7-*(8$("9f(0). !"' #4(,:'#;"+&+"9'*(+
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(2),
! =1
2!
1
3 ![5f4(0)+ 3f2(0)+1]=
1
6 !(5 !04 + 3 !02 +1)" ! =
1
6.
10. f:!! ! ,
xex![f(x)+ 2]= ex ![f(x) !!x+ex]"1 , x! !* .
xex!f(x)+ 2xex= ex !f(x)!!x+e2x"1# xex !f(x)"ex !f(x) !!x= e2x"1"2xex#
!f(x) "(xex#e
x"!x)= e2x#1#2xex .
xex!ex "!x# 0 , x! 0 , f(x)=e2x!1!2xex
xe
x!
e
x"
!
,
!* .
, f ! , 0,
! imx!0
f(x)=f(0) .
, ! =! imx!0
e2x"1"2xex
xex
"ex
#!x=
0/0
! imx!0
(e2x"1"2xex $)
(xex"ex #!x $)%
! ! = ! imx"
0
e2x#(2x $) %(2x $) #ex%2x#(ex $)
(x $) #ex
+ x#(ex
$) %(ex
$) #!x%ex
#(!x $)
= ! imx"
0
2e2x%2ex%2xex
e
x
+ xex
%ex
#!x%ex
#"#$x
!
! ! = ! imx"0
ex(2ex#2#2x)
ex(1+ x# !x#"#$x)
= ! imx"0
2ex#2#2x
1+ x#!x#"#$x!
0/0
! ! = ! im
x"0
(2ex#2#2x $)
(1+ x# !x#"#$x $)= ! im
x"0
2ex#2
1#"#$x+ !x=
0/0
! imx"0
(2ex#2 $)
(1#"#$x+ !x $)!
! ! = ! im
x"0
2ex
!x+ "#$x=
2e0
!0+ "#$0! ! = ! im
x"0f(x)= 2=f(0) .
f(x)=
e2x!1!2xex
xex!ex"!x
, "# x# 0
2 , "# x= 0
$
%
&&&&
'
&&&&
.
!"#$%"&'($. )*+,-.*/0*&, 123"4"$#&)23"2$516(.
789:;:'3.!"#$#%&'()de l' Hospital.
)23"2$51-+$
