Oria Synexeia G Lykeiou
36
ΦΡΟΝΤΙΣΤΗΡΙΑ ΗΛΙΑΣΚΟΣ ΥΠΗΡΕΣΙΕΣ ΠΑΙΔΕΙΑΣ ΥΨΗΛΟΥ ΕΠΙΠΕΔΟΥ Θετικής - Τεχνολογικής Κατεύθυνσης Μαθηματικά Γ΄Λυκείου Όρια - Συνέχεια ΣΤΕΦΑΝΟΣ ΗΛΙΑΣΚΟΣ e-mail: [email protected] www.iliaskos.gr
description
Functions, limits, Continuity
Transcript of Oria Synexeia G Lykeiou
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e-mail: [email protected] www.iliaskos.gr
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73
3. 3.1
x0 .
lim f xx x0
,="
] g
x0 - x0 x0 -
x R0 ! lim f xx x0 ,=" ] g ,x0 .
lim f xx x0" -
] gx0 x x< 0).
lim f xx x0" +
] gx0 x x> 0).
x0
lim lim limf x f x f xx x x x x x0 0 0
= =" " "- +
] ] ]g g g3.2 x R0 !
x0 .
lim f x 0>
x x0"] g f x 0>] g x0 .
lim f x 0x x0" ] g lim f x 0
x x0H
"] g .
f x 01] g x0 lim f x 0a lim
xx 3a =+
" 3+ lim 0
xxa =
" 3-.
0 1< a limlog xx
3=+" 3
a+ limlog x
x 03=-
"a+ .
0 1<
-
83
lim x x x15 2 1
x 2$3 3- + - =- - =+
" 3-b ]l g .
lim x x x5 2x
2 + - +" 3+
^ h.lim limx x x x x x x5 2 1
5 2x x
2 22+ - + = + - + =
" "3 3+ +^ bch l m
lim limx x x x x x x x15 2 1 5 2
x x2 2+ - + = + - + =
" "3 3+ +bc bcl m l m
lim x x x15 2 1 2
x 2$3 3+ - + =+ =+
" 3+bc l m; E .
lim x x x5 2x
2 + - +" 3-
^ h.-0 $3lim limx x x
x x xx x x5 2
5 25 2
x x2
2
2 2
+ - + =+ - -
+ - =-" "3 3- -
^ h
lim limx x x x
x
x x x x
x x
1 5 25 2
1 5 25 2
x x
2 2+ - -=
- + - -=-
-" "3 3- -bc b
bl m l
l
limx x x
x x
1 5 2 1
5 2
1 0 0 15 0
25
x
2- + - +=
- + - +=-
- -" 3+ bc
b^l m
lh .
x " 3+ -
x " 3- -
x " 3+ -
x " 3- -
lim e
e2 1
3x x
x
++
" 3+.
lim lim limee
e e
e e
e
e2 1
32 11 3
2 11 3
21
x x
x
x xx
xx
xx
x
++ =
+
+=
+
+=
" " "3 3 3+ + +``
jj
.
-
84
lim ee2 2
3x x
x
++
" 3-.
lim e
e2 2
32 00 3
23
x x
x
++ = +
+ =" 3-
.-
x " 3+
lim 0
xxa =
" 3+0 1< a .
lim ee2 3
5 2x x x
x x$++
" 3+.
lim limee
e e
e e2 3
5 22 3
1 5 2
x x x
x x
x xx
xx
$++ =
+
+=
" "3 3+ + ````
j jj j
lime
e2 3
1 5 2
31
x x
x
+
+=
" 3+ ``
jj
.
lim ee2 3
4 2x x x
x x
1$
++
" 3- +.
lim lim limee
ee
e
e
2 34 2
2 2 34 2
2 2 3 2
2 2 4x x x
x x
x x x
x x
x x x
x x
1$
$$
++ = +
+ =+
+=
" " "3 3 3- + - - ````
j jj j
lim e
e
2 3 2
2 424 2
x x
x
+
+= =
" 3- ``
jj
.
lim lnlnxx1x +" 3+ .
lim ln
ln limln ln
ln limln
xx
x x
x
x1 1 1 1 1
11 01 1
x x x+ = +=
+= + =" " "3 3 3+ + +` j .
- x 1Ghn x 1Gvyo
-
lim xx
x
hn" 3+
.
-
85
xx
x x xx
x1 1 1+G G Ghn hn- .
lim x1 0
x- =
" 3+c m lim x1 0x =" 3+
lim xx 0
x
hn =" 3+
. hn - lim x
1 0x
=" 3+
lim xx 0
x
hn =" 3+
-
3.6
f x] gx0 f x] gx0 lim f x f xx x 00 =" ] ]g g.
f x] g
P x] g. Q x
P x]] gg hn vyo f{ v{ . a log xa 0 1< !a .
x0
x0 f g+ c f$ f g$ g
f fvx0 .
x0
f x0] gg f% x0 .
--
-
-
86
x0
-x1 , x0 . -
lim f xx x1"
] g-x1 .
3.7
- ,a b^ h ,a b^ h.
- ,a b6 @ ,a b^ hlim f x fx
a="a+
] ]g g lim f x fx
b="b-
] ^g h.
,a b6 @ f f 0
-
87
---
,a b^ h
lim limf x f x 0] g R! lim f xx 0" +
] g- lim f x 0>
x 0" +] g .
lim f x 0x 0
=" +
] g lim f x1x 0 3=+" + ] g . lim f x 0
x x0=
" +] g lim f x1x x0 3=+" + ] g .
lim f xx 0
2 3=+"
^ h lim f xx 0
3=+"
^ h lim f xx 0
3=-"
^ h . lim f x g x
x 0$
"^ ^^ h hhf g0 0$^ ^h h.
f x 0
-
,a b6 @ f x 00 =^ h ,x0 ! a b^ hf f 0
-
lim x xx3 2
4x 2 3 - +
-"
limx
x x11 1
x 1 2 -+ - -
".
lim x xx x x
2 1 12 3 1 2
x 0 2
2
- + - -- - - + -
" lim x
x x x4
8 16 4x 4
2
-- + - -
".
lim xx x x
12 3 1
x 1 2
2
-- + - +
" lim x x
x x3 2
2 3 12x 2 2
2
- ++ + -
".
lim x xx x
7 62 2
x 1 2 - +- + -
" lim x
x x x9
1 14x 3 2
2
-- + + -
".
lim x
x x x1
1 3x 1 2
2
-- + +
"- lim x
x x1
1 1x 1
2
-- + -
".
lim xx x x
24 5 6
x 2
2 2
-- + - +
" lim x
x x1
2 1x 1
2
-- +
".
lim xx x x
11 3 2
x 1 2
2
-- + - +
" lim x x
x x5 6
2 4x 2 2
2
- +- + -
".
lim x xx x
9 82 2 4 7
x 1 2 + ++ + - -
"- lim x x
x x x2 7 4
6 4 1 4 2x 4 2 + -
+ + + + +"-
.
lim x xx x
22 1 1
x 2 2 ++ +- +
"- lim x x
x x3
3 3x 3 2 -
- + -"
.
lim x x xx x3
x 0 2
3
++ -
" lim x
x x x1
10 9 1x 1
2
-- + - -
".
lim x
x x x2x 0
2 + -"
lim x xx x5 4
4 2 8x 4 2 - +
- + -"
.
lim xx x x
93 3
x 3 2
2
-- + -
" lim x x
x x1 2 3x 1 2
3
+ --
".
lim xx x x x
22 5 6
x 2
2 2
-- - - +
" lim x x
x x x7 6
2 1 2 2x 2 2
3
- +- + + -
".
lim x
x23
x 0
hn"
lim xx
42
x 0 f{f{
" lim x
x2x 2 rvyo
-" r .
lim x xx x
x 0 hnhn
+-
" lim x
x x1 3 1 2x 0
hn hn+ - -"
.
limxx
3 9x 0hn- +"
.
lim x
xx 0
f{"
lim xx x
x 0
2
hn-
" lim x
x x3x 0
2
f{+
".
-
93
lim x xx x
42
x 0 2f{ hn
+-
" lim x
xx 0
hn"
lim xx
x 0
hn"
.
lim xx1 1
x 0
hn- +"
limx
x1 2x 0 vyo-"
.
lim x xx x x
x 0 2
4 2 hn+
+ -"
lim x xx x
5 73 4
x 0 hnhn
+-
".
limx x
x3 3
1 2x 0
vyo+ -
-" ^ h lim x x1x 0 2vyo-" .
,
,f x
x x xx x
3 5 42 4pi
>2
Gvyo=
- --
^ ^h h) x 40 = .
,
,f x x x
x x x
xx x x x
3 26 8 2
24 4 2 2
>
2
2
Ga
b=
+ +- +
^ h ) lim f xx 2"
^ h- ,M 1 3-^ h.
, R!a b , ,
, ,, ,
f xx xx x xx x
1 12 1 43 2 4 5
3
2
!
!
!
a
a b
b
=+ -+ ++
^ ^^
h6 @
@@
Z
[
\
]]
] lim f x
x 1"^ h
lim f xx 4"
^ h. , R!a b lim x
x x4 2x 2 2
2 a b-
- + ="
.
-
94
lim f xx x0"
^ h lim f x x4 3 1
x 2+ - =
"^^ h h lim xf x 32 1 4x 3 - - =" ^ h .
limx x
f x1 1
2x 3 + - +
="
^ h lim xf x 43 5 5x 4 - - =" ^ h . x f x x3$ Hhn hn^ h R! lim f x R
x 0!l=
"^ h -
l. f x^ hx0
xx f x x x
x24
212 242
2G G--
- --^ h x 20 = .
xx f x x x
x11
4 31
2G G--
- +-^ h x 10 = .
x xx x x f x x x
x2 3
4 33 10 32 9
2
3 2
2
2
G G+ -+ +
+ +-^ ^h h x 30 =- .
xx f x x
x1 2 1G Ghnvyo vyo- -^ h x 00 = .
x xf x xG Ghn f{^ h x 00 = . x f x x4 42G G +^ h R!
lim xf x
28
x 2 --
"
^ h lim xf x x22x 22
--
"
^ h.
f x^ h x f x f x x4 5 2 2 32 2G G- - -^ ^h h R! lim f x
x 1"^ h.
lim g xf x
Rx x0
!a=" ^^ hh f x g x f x g x3 43 2 3+ =^ ^ ^ ^h h h h
R! . lim f x 0
x 2=
"^ h
limf x
f x1 1x 2
hn
+ -" ^^^
hhh lim f xf x f x2 22 1 1x 2
2
- -- + -
" ^^ h^h h . lim f x x x2 4 2 5
x 3
2+ - - ="
^^ h h lim f xx 3"
^ h. x f x f xa+ = -^ ^h h R! lim f x x2 1
xa+ - =
"a^^ h h
lim f xx 0"
^ h. lim x
f x 1x 0
="
^ h limx
f x11
x 1 --
"
^ h.
lim lnf x x x1 3 3x 1
2- + - + ="
^^ h h lim f xf x1x 0 2hn r -" ^ ^^ h hh . lim x
f x x 1 2x 0
hn+ -=
"
^ h limf x xf x 1
x 0 vyo--
" ^^hh
.
-
95
f x f x x x x1 2 4 22- + = - +^ ^^h h h R! lim x
f x f11
x 1l-
- ="
^ ^h h 0>l . lim x
f x4 3
1x 2 2 - ="
^ h lim g x x 8 32x 2 3 - =-" ^ ^^ h hh limh xx 2" ^ hh x f x g x=^ ^ ^h h h.
lim x f x1 1 4x 0
+ - ="
^ ^_ h hi lim g x x 1 2x 0
vyo - ="
^ ^^ h hh lim g x
f xx 0" ^^ hh .
lim f x g x3 19x 3
+ ="
^ ^^ h hh lim f x g x2 3x 3
- =-"
^ ^^ h hh lim f xx 3"
^ h limg xx 3"
^ h.
lim xx x
48 16
x 4 4
2
-- +
" ^ h lim x xx8 163x 4 2 - +-" lim x x
x2 1
2 3x 1 2 - +
+"
limx x x
x3 3 1
1x 1 3 2
2
+ + +-
"-.
limx
x xx 0 54
$hn vyo"
limxx
x 0 43
hn"
.
lim x x x
x3 3 15 6
x 1 3 2- + --
" lim x x x
x4
22x 2 2 2- - -" a k.
lim xx
x x1 22
x 1 2 2- - + -" a k lim xx x x12 1 1x 1 -- - -" c m. lim x x
1 1x 0
-"
c m lim x xx x3 5x 02 + -
".
lim
x x xx
11
x 1 3 2- - +-
" lim
x xx
25
x 4 - --
".
limx x x x2
32x 4 - +
--"
lim x x21
21
x 2 - - -" c m. lim
xx
13 5
x 1 --
" lim x
x1x 0 2
vyo-"
.
lim x
x1
2x 2
hn --
" r lim x
x 1x 0 hn
-"
.
lim xx
13
x vyo +-
"r lim x
x1
1x 0 vyo -
-"
.
-
96
lim xx
2 11
x 3vyo -
-" r
lim xx
2 13
x 67 hn +
-" r
.
R!a lim x x x x
x x4 6 4 1
2x 1 4 3 2
2 2a a- + - +
- +"
lim xx x
22
x 2
2 a a-
- +"
.
lim x x xx x3 3 1
5x 1 3 2
2 a- + -
+ -"
, R!a b -
lim x xx x2
x 0
2
hna b+ +
" lim x x
x x4 4
2 4x 2 2
2 b a- ++ +
".
lim xx x 1
x 0 2a b+ - +
"
lim f x
x 1"^ h lim xx f x11x 1 3 3-- =+" ^ ^h h .
lim f xx 0"
^ h lim f x x x2 3x 0 3hnf{ =+" ^ h . lim f x
x 4"^ hx 4! f x x x xx x12 48 645 43 2
2
H - + -- +^ h .
lim f xx 1"
^ hx 1! f x x x xx x3 3 15 43 22
G - + -- +^ h .
lim f xx 3"
^ hx 3! f x xx 31 2H -+ -^ h .
lim x x x3 2 5x
3 2- + -" 3+
^ h lim x x x3 5 2 5x
3 4- + -" 3+
^ h. lim x x x3 2 7
x4 3- + -
" 3-^ h lim x x2 4 5
x3 2- +
" 3-^ h.
lim x x x2 5 6x
3 2+ - -" 3-
lim x x x2 3 7x
5 2- + +" 3-
^ h
lim x xx x2 1
5 2x 3
4 2
+ -- +
" 3- lim x
x x1
4 4 6x 2
3 2
-- +
" 3+.
lim x xx x
6 3 112 5 6
x 6
6 2
+ -+ -
" 3- lim x x
x x x2 1
4 2x 5 3
5 3
- +- + -
" 3-.
lim x xx x
3 19 16 7
x 5 2
3
- -+ -
" 3- lim x
x x2 7
4 5 2x 100
99
++ +
" 3-.
lim x xx x x x x
4 5 82 5 1 7
x 2
3 2 3
- ++ - + + -
" 3- lim x
xx
x x11
2 3x
2 2- - +- -
" 3+c m.
-
97
lim x x2 1x
23 + -" 3+
lim x x7 5 3x
2 + -" 3-
.
lim x x x x2 4 2 1x
2 5 25+ - + - + +" 3-
^ h. lim x x x1
x2 - + -
" 3-^ h.
lim x x x2x
2 + +" 3-
^ h lim xx x2 54 4x2
-+ +
" 3-.
lim x x x8 1x
33 + - -" 3+
^ h lim x x x5 3x
4 + - +" 3-
^ h. lim x x x4 1 2
x2 + - -
" 3+^ h lim x x x1
x2 + + -
" 3+^ h.
lim x x x1
x2 + + +
" 3-^ h lim x x x9 2 3
x2 + + +
" 3-^ h.
lim x x x1x
2 2+ +" 3-
^ h lim x x x x1 4x
2 2+ + - -" 3-
^ h. lim x x x x4 1 1 3
x2 2+ + + + -
" 3+^ h.
lim x x x x9 2 1 1 2x
2 33+ + - + -" 3+
^ h.
limx x xx x x
1 11 1
x 2 2
4 4
+ - + ++ + - +
" 3- lim x x 1
x3 3- -
" 3-^ h.
limxx
xx11x
2 +- +
-" 3+
lim x x x xx
+ - -" 3+
_ i. lim x x x x
x+ + -
" 3+_ i.
lim x x x x x4 1 3x
2 2+ + + + -" 3+
^ h.
lim ee e3 2
1x x
x x2
++ -
" 3+
+
lim 1 2 66 3 6 3
x x
x x
2
1
$$
-- +
" 3+ +
+
.
lim 2 52 1
x x
x 1
+-
" 3+
+
lim 2 2 12 2 3
x x x
x x
3
1 2
+ -+ -
" 3- +
- +
.
lim e ee e
12
x x x
x x
1 1
2
+ -+ -
" 3+ - +
+
limee
31
x x
x
01
1
++
" +.
lim e
e33
x x x
x x
1
1
+-
" 3+ +
+
lim ee
22
x x x
x x
2
1 2
++
" 3+ +
+ +
.
lim lnlnxx1x +" 3+ lim ln
lnxx1x +" 3- .
lim ee
3 22 3
x x x x
x x x
1 1 2
2 1
+ -- +
" 3+ + - -
+ -
lim ee
3 22 3
x x x x
x x x
1 1 2
2 1
+ -- +
" 3- + - -
+ -
.
-
98
lim x x 5
xx6 + -
" 3-^ h lim x2 3
xx x 6- -
" 3+-^ h.
lim x xe
12
x
x
2+ ++
" 3- limln x x1
x+ -
" 3+^ h.
limln x xex
x 0 2hn+" a k lim 1 21x x0 1+" + .
lim x
xx
hn" 3+
lim xx
x
vyo" 3-
.
limx x1
xhn
" 3+ limx x
1x 0
hn"
.
lim x xx x x3 2
2 3x
2
vyohn
+-
" 3- lim x
xx1
x 0
2
hn hn" .
R!a
lim x xx x
21
3 72 3
x 2
3
aa--
+ -- +
" 3+ ^^ hh lim x x xx x x4 5 2 8x4 2
3 2aa-
- + -+ - +" 3- ^ h .
lim x x x3 2 1x
2 a+ + - +" 3-
^ h lim x x x3x
2 a+ + +" 3-
^ h. 0>a
lim 3 13 1
x x x
x x
1
1
aa
+ -+ +
" 3+ -
+
lim 22 4
x x x
x x
1
1
aa+
-" 3+ -
+
.
, R!a b
lim x xx x
11
x 3
2
a ba a b
- +- - +
" 3+
^ h lim x xx x31 2 1x 22
ba
+ -- + -
" 3+
^ h.
: ,f 1 R"3+^ h xx x f x x
x2
9 3 1 3 42 G G+- + +^ h ,x 1 3! +^ h.
lim f xx" 3+
^ h. R
xx f x x
x11
13
2
2
1 1++
++^ h x 0> .
lim f xx" 3-
^ h. lim x
P x2 2x + =" 3+
^ h lim xxP 2x 0 hn l=" ^ h - R!l .
:f R R" lim f x x x x2 1 3x
2+ - + + =" 3+
^_ h i , lim f xx" 3-
^ h. x f x g x f x g x22 2 H+ -^ ^^ ^ ^^h hh h hh R!
lim f xx" 3-
^ h limg xx" 3-
^ h.
-
99
xf x
x x1 1
2Ghn-^ h R! lim f x
x" 3+^ h.
R x f x x x1 3 2 G+ -^ ^h h x 0> lim f x
x" 3+^ h.
f x e2 1x xH+^ ^h h R! lim f xx" 3+
^ h lim f xf x f xf x2 3 52 1x 33 2
+ -- +
" 3+ ^ ^^ ^h hh h . f x x x 14 G vyo -^ h R!
lim f x f x f x2 3x 0
2 - + +"
^ ^ ^_ h h hi. f x x x exHhn^ h R!
lim f x f x f x1x 0
2 + + -"
^ ^ ^_ h h hi. lim f x
f xx x0
3ba
=+--
" ^^ hh , R!a b lim f xx x0" ^ h. lim x
f x x1x 20
3vyo-
=-"
^ ^h h lim f xx 0"
^ h. lim x
f x x 1x 1
3hn r
=--
" ^^ ^ hh h lim f xx 1" ^ h. lim x
f x e25
x
x x1 2
hn =+
" 3+
+ -^ h lim f xx" 3+
^ h. f x f x2= -^ ^h h R!
lim f x x x1 2x
3 3- + + =" 3+
^_ h i , lim f xx" 3-
^ h. f x xf x x
x2 1 11- = -
+^ `h j x 1R! - " , lim f x
x" 3+^ h lim f x
x 1"^ h.
lim f xx X0
3=+"
^ h lim ln x f x2 3x x
f x 20
+ + -"
^ ^^ h hh6 @. lim x
xf x
2 02x 2 hn r
+ =-" ^ ^h h f x 0^ h R!
lim f xf x f x
2 13 2
x 0
2
-+ -
" ^^ ^hh h .
-
,
,
,
f xx xx x x
x
xx x x
0
1 0
0
>
!=
+
=^ ^h h* .
,,
f x e xx x1 0
032
Gvyo r= -
- + -^ ^h h* .
,
,
,
xfxx x
x
xx x
11 1
21 1
13 2 1
>
x
2
2
Ga= + -
- + -
b a-^ h * .
-
, R!a b -
,
,
,
f xx
xx
x
xx x
2 0
21 0
0
>
2 Ga=
-
+ -^ ^ ^^h
h hh* .
f x f y x yG- -^ ^h h ,x y R! R .
lim xx f x x
11
2x 1
hn r-
- -=
"
^ ^ ^h h h f 1^ h.
f x x2Hvyo r-^ h ,x 0! r6 @-R f 2
r` j. f 0 4=^ h f x x x f x1 22 2hn+ = +^ ^h h x R*!
f x x f x3 2 hn= +^ ^h h R!
f x xG^ h R! . x 00 = .
xf x x f x x x1 2 12Ghn- - - + +^ ^ ^h h h R! - ,A 1 1^ h.
x 00 = xf x x1 5< vyo+^ h x R*! f 0^ h.
f x f x2 - =^ ^h h R!
f x f x x x1 12+ + = + +^ ^h h R!
f x y f x f y+ =^ ^ ^h h h ,x y R! R .
f x y e f y e f x xyx y+ = + +^ ^ ^h h h ,x y R! -R .
x x2 3 0hn vyo+ = ,0 r^ h. x x 13 = + ,1 2^ h. x x 4vyo+ = ,2r r^ h. x
xxx1 1 0
2 6
a b-+ + -
+ = ,a b^ h.
-
xx
xx
11
23 0
10 8
-+ + -
+ = ,1 2^ h. x
xx
x4 3 0rf{
rv{
- + - = ,4 3r r` j.
,a b6 @ f gb a=^ ^h h f g g!a b a=^ ^ ^h h h.- ,!t a b^ hf gt t=^ ^h h.
R f x g x0G G^ ^h h R! , R!a b f a a=^ h g b b=^ h
R!c f gc c c+ =^ ^h h . ,0 16 @f 0 a=^ h f 1 b=^ h
, ,0 1!a b ^ h f x x=^ h - ,0 1^ h.
: , ,f "a a a a- -6 6@ @ 0>a Cf y x=- .
: ,f R"a b6 @ f 2a b=_ i f 2b a=^ h . ,!i a b^ h f 2i i=^ h , R!a b
-
: ,f R"a b6 @ f f 3
2a
a b= +_ ci m ,!i a bh6 f f 3i i
b a= + -^ ch m, , R!a b ,a b6 @
,a b6 @ , ,f g+ 4!a b a b^ ^h h6 6@ @ ,
,!i a b6 @f gi i=^ ^h h.
f x x x x3 2a b ab ab= - + - +_ ^i h , , R!a b ,0 1!i 7 A f 2
1 1i
a b= - -^ ^ ^h h h . x x16 116 - =
R . x x2 2avyo b+ =
:f RR " f x x x3 2a b= + +_ i 0>b f x 0=_ i
, :f g RR " f x g x xl- =_ _i i
R! - 'x x - f x 0=_ i R .
R , R3a b6 @ ,a b6 @ f f x f x f 02a l b+ + =_ _ _ ^i i i h R .
, : ,f g R"a b6 @ , R!a b , ,f f ga b b a b b= = =_ ^ ^i h h g a a=_ i . ,!i a b^ h f g g f 2% %i i i+ =_ ^ _ ^i h i h .
-
,a a-8 B R*!a
g a a=-_ i g a a- =_ i . ,x0 ! a a-_ if g x f x g x 00 0 0+ + =__ _ _ii i i .
, : ,f g R"a b6 @ , , ,f ga b a b a b= =^ ^h h6 6 6@ @ @. ,!i a b6 @
f g g f 2i i i+ =^^ ^^hh hh .
, , : , ,f g h "a b a b6 6@ @ , R!a b
, , ,f ga b a b a b= =^ ^h h6 6 6@ @ @. h a b=_ i h b a=^ h . ,!i a b^ h f g h g fi i=^^^ ^^hhh hh.
x x xf x x x2 8 4 6
3 2 6G G hn+ - + +_ i ,x 4 3! - + i7 . f 0 6
1=^ h . ,0 1!l _ A
f 66l hn l l= +_ i .
R -
lim xf x
1 4x 1 - ="_ i
x x f x x4 2 2 42G Ghn - - -^ ^ _h h i R! .cf -y x x 12= - + ,1 2_ i.
,0 16 @ - f 0 2=^ h f 1 4=^ h xf
f f f f4
51
52
53
54
=+ + +^ ` ` ` `h j j j j
,0 1^ h. ,a b6 @ , R!a b
-
,a b6 @ , R!a b ,!i a b6 @
f f f f91 2 3 2 4i a
a bb= + + +^ _ c ^h i m h< F.
,a b6 @g x 0!_ i ,x ! a b6 @ ,!p a b^ h g
f 1 1pp
p a p b= - + -^^ hh . ,a b6 @ , ,1 2 !t t a b^ h
f x 0=_ i ,1 2t t` j
,x 1 0! -6 @x xf x2 83 2+ =^ h
,x ! a b6 @ f x 0!^ h -
R z x if x= + _ i R! ,f f2 2 23 2a a a a= =-c cm m ,x 0 20 ! a_ i ,z z2< Ha a a a- -
z2 3 Ha a- ,f 0a =_ i .
f x x x 4vyo= - -^ h ,x 2! r r^ h. xf e 1x= -^ h ,x 0 3! + h6 . lnf x e x1 1x= - + +^ ^h h ,x 0 3! + h6 . lnf x x x=^ h ,x 0 3! +^ h.
:f R R" f 0 0=^ h lim f xx
3=+" 3-
^ h x 0
