Όρια - Συνέχεια...
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Μαθηματικά Γ΄ ΛυκείουΘετικού Προσανατολισμού
Όρια - Συνέχεια
ΘΕΩΡΙΑ - ΜΕΘΟΔΟΛΟΓΙΑ - ΑΣΚΗΣΕΙΣ
2
3. ɄʌɿɲͲɇʐʆɹʖɸɿɲ3.1 ȸɹʆʆʉɿɲʏʉʐʉʌʀʉʐɃʌɿʍʅʊʎ Ʉʌɿʉɸʀʆɲɿɻʏɿʅɼ ʋʉʐʋɲʀʌʆɸɿɻʍʐʆɳʌʏɻʍɻĨʃɲɽʙʎʏʉdžʋʄɻʍɿɳɺɸɿ ʅɸʉʋʉɿʉʆɷɼʋʉʏɸʏʌʊʋʉʏʉx0 .
lim f xx x0
,="] g
Ʌɲʌɲʏɻʌɼʍɸɿʎͻ ȳɿɲʆɲɹʖɸɿʆʊɻʅɲɻɲʆɲɺɼʏɻʍɻʏʉʐʉʌʀʉʐʏɻʎĨʃʉʆʏɳʍʏʉx0 ʋʌɹ-
ʋɸɿɻ Ĩ ʆɲʉʌʀɺɸʏɲɿ ʃʉʆʏɳʍʏʉ x0 ɷɻʄɲɷɼ ʏʉ x0 ʋʌɹʋɸɿ ʆɲ ɸʀʆɲɿɸʍʘʏɸʌɿʃʊʍɻʅɸʀʉʏʉʐʋɸɷʀʉʐʉʌɿʍʅʉʑʏɻʎĨɼɳʃʌʉɲʆʉɿʖʏʉʑɷɿɲ-ʍʏɼʅɲʏʉʎʏʉʐʋɸɷʀʉʐʉʌɿʍʅʉʑʏɻʎĨ
ͻ ɉʋɳʌʖɸɿʏʉʊʌɿʉʏɻʎĨʍʏʉx R0 ! ʃɲɿʏʉʍʐʅɴʉʄʀɺʉʐʅɸʅɸlim f xx x0
,="] g
ʊʏɲʆʉɿʏɿʅɹʎʏɻʎĨɴʌʀʍʃʉʆʏɲɿʊʍʉʃʉʆʏɳɽɹʄʉʐʅɸʍʏʉʆɲʌɿɽʅʊ,ʃɲɽʙʎʏʉdžʋʌʉʍɸɶɶʀɺɸɿʏʉx0 .
ɃʌɿʍʅʊʎȺɲʄɹʅɸʋʄɸʐʌɿʃʊʊʌɿʉʏɻʎĨɲʋʊɲʌɿʍʏɸʌɳʃɲɿɽɲʏʉʍʐʅɴʉʄʀɺʉʐʅɸ
lim f xx x0" -] gʊʏɲʆʏʉdžʋʄɻʍɿɳɺɸɿʏʉx0 ʅɸʅɿʃʌʊʏɸʌɸʎʏɿʅɹʎ;x x< 0).
ɃʌɿʍʅʊʎȺɲʄɹʅɸʋʄɸʐʌɿʃʊʊʌɿʉʏɻʎĨɲʋʊɷɸʇɿɳʃɲɿɽɲʏʉʍʐʅɴʉʄʀɺʉʐʅɸ
lim f xx x0" +] gʊʏɲʆʏʉdžʋʄɻʍɿɳɺɸɿʏʉx0 ʅɸʅɸɶɲʄʑʏɸʌɸʎʏɿʅɹʎ;x x> 0).
Ʌɲʌɲʏɼʌɻʍɻȳɿɲʆɲʐʋɳʌʖɸɿʏʉʊʌɿʉʏɻʎ Ĩʍʏʉ x0 ɽɲʋʌɹʋɸɿʆɲʐʋɳʌʖʉʐʆʏɲ
ʋʄɸʐʌɿʃɳʊʌɿɲʃɲɿʆɲɸʀʆɲɿʀʍɲlim lim limf x f x f xx x x x x x0 0 0
= =" " "- +] ] ]g g g
3.2 Ʉʌɿʉɇʐʆɳʌʏɻʍɻʎʍʏʉx R0 !
Ȼɷɿʊʏɻʏɸʎʉʌʀʘʆɇɸʊʄɸʎʏɿʎʋɲʌɲʃɳʏʘɿɷɿʊʏɻʏɸʎʐʋʉɽɹʏʉʐʅɸʊʏɿʐʋɳʌʖʉʐʆʏɲʊʌɿɲ
ʍʏʉx0 .ͻ Ʉʌɿʉʃɲɿɷɿɳʏɲʇɻ
ͻ Ȱʆ lim f x 0>x x0"] g ʏʊʏɸf x 0>] g ʃʉʆʏɳʍʏʉx0 .
ͻ Ȱʆ lim f x 0<x x0"] g ʏʊʏɸf x 0<] g ʃʉʆʏɳʍʏʉx0 .
Ʌʌʉʍʉʖɼ Ȱʆ lim f x 0x x0
H"] g ʏʊʏɸɷɸʆɿʍʖʑɸɿʊʏɿf x 0H] g ʃʉʆʏɳʍʏʉx0 .
,
3
Ȱʆʏʀʍʏʉɿʖɲɲʆlim f x 0x x0
G"] g ʏʊʏɸɷɸʆɿʍʖʑɸɿʊʏɿf x 0G] g ʃʉʆʏɳ
ʍʏʉx0 .ͻ Ȱʆf x 0H] g ʃʉʆʏɳʍʏʉx0 ʏʊʏɸ lim f x 0
x x0H
"] g .
ͻ Ȱʆf x 0G] g ʃʉʆʏɳʍʏʉx0 ʏʊʏɸ lim f x 0x x0
G"] g .
Ʌʌʉʍʉʖɼ Ȱʆ f x 02] g ʃʉʆʏɳʍʏʉ x0 ʏʊʏɸɷɸʆ ɿʍʖʑɸɿʊʏɿ lim f x 0>x x0"] g
Ȼʍʖʑɸɿʊʅʘʎ lim f x 0x x0
H"] g .
Ȱʆʏʀʍʏʉɿʖɲ ɲʆ f x 01] g ʃʉʆʏɳ ʍʏʉ x0 ʏʊʏɸ ɷɸʆ ɿʍʖʐɸɿ ʊʏɿlim f x 0<x x0"] g Ȼʍʖʑɸɿʊʅʘʎ lim f x 0
x x0G
"] g .
ͻ Ȱʆf x g xG] ]g gʃʉʆʏɳʍʏʉx0 ʏʊʏɸʃɲɿ lim limf x g xx x x x0 0
G" "] ]g g.
Ʌʌʉʍʉʖɼ Ȱʆ f x g x<] ]g g ʃʉʆʏɳ ʍʏʉ x0 ʏʊʏɸ ɷɸʆ ɿʍʖʑɸɿ ʊʏɿlim limf x g x<x x x x0 0" "] ]g g.Ȼʍʖʑɸɿʊʅʘʎ lim limf x g x
x x x x0 0G
" "] ]g g.
ͻ Ȱʆ lim limf x g xx x x x0 0
G" "] ]g gʏʊʏɸʃɲɿf x g xG] ]g gʃʉʆʏɳʍʏʉx0 .
Ʌʌʉʍʉʖɼ Ȱʆ lim limf x g x<x x x x0 0" "] ]g g ʏʊʏɸ ɷɸʆ ɿʍʖʑɸɿ ʊʏɿ f x g x<] ]g g ʃʉʆʏɳ
ʍʏʉx0 .ͻ ȾʌɿʏɼʌɿʉɅɲʌɸʅɴʉʄɼʎ;ʃʌɿʏɼʌɿʉʍɳʆʏʉʐɿʏʎͿ
ͻ Ȱʆ ɿʍʖʑɸɿ h x f x g xG G] ] ]g g g ʃʉʆʏɳ ʍʏʉ x0 ʃɲɿlim limh x g xx x x x0 0
,= =" "] ]g g ʏʊʏɸʃɲɿ lim f x
x x0,=
"] g .
ͻ Ʉʌɿɲʃɲɿʋʌɳʇɸɿʎͻ lim lim limf x g x f x g x
x x x x x x0 0 0! !=
" " "] ]^ ] ]g gh g g.
ͻ lim limf x f xx x x x0 0l l=
" "]^ ]gh g.
ͻ lim lim limf x g x f x g xx x x x x x0 0 0
$ $=" " "] ]^ ] ]g gh g g.
ͻ lim limlim
g xf x
g xf x
x xx x
x x
00
0="
"
"]] ]]
gg ggʅɸ limg x 0
x x0!
"] g .
ͻ lim limf x f xx x
vx x
v
0 0=
" "]^ ]_gh gi .
ͻ lim limf x f xx x x x0 0
=" "] ]g g .
ͻ lim limf x f xx x x x0 0
=" "
l l] ]g gɲʆf x 0H] g ʃʉʆʏɳʍʏʉx0 .ͻ Ʉʌɿʉʋʉʄʐʘʆʐʅɿʃɼʎ
ͻ limP x P xx x 0
0=
"] ]g g.
ͻ Ʉʌɿʉʌɻʏɼʎ
ͻ lim Q xP x
Q xP x
x x 0
0
0=
" ]] ]]gg gg ,ʅɸQ x 00 !] g .
4
ͻ Ɉʌɿɶʘʆʉʅɸʏʌɿʃɳʊʌɿɲͻ lim x x
x x 00hn hn=
".
ͻ lim x xx x 0
0vyo vyo=
".
ͻ lim xx 1
x 0
hn ="
.
ͻ lim xx 1 0
x 0
vyo - ="
.ͻ Ʉʌɿʉʍʑʆɽɸʏɻʎʍʐʆɳʌʏɻʍɻʎ
ͻ lim limf g x f ux x u u0 0
=" "
]^ ]gh g,ʊʋʉʐu g x= ] g.Ʌʌʉʍʉʖɼ ȴɸʆʍʋɳʅɸʋʉʏɹʏʉʊʌɿʉʍɸɸʋɿʅɹʌʉʐʎʊʌɿɲʖʌɻʍɿʅʉʋʉɿʙʆʏɲʎ
ʏɿʎɿɷɿʊʏɻʏɸʎɲʆɷɸʆɶʆʘʌʀɺʉʐʅɸʊʏɿʊʄɲʏɲɸʋɿʅɹʌʉʐʎʊʌɿɲʐʋɳʌʖʉʐʆ
Ʌʌʉʍʉʖɼ ȵʀʆɲɿɷʐʆɲʏʊʆʆɲʐʋɳʌʖɸɿɹʆɲʊʌɿʉʖʘʌʀʎʆɲʐʋɳʌʖɸɿʃɳʋʉɿʉɲʋʊʏɲɸʋɿʅɹʌʉʐʎʊʌɿɲ
Ⱥɸʙʌɻʅɲ;ɀɻɷɸʆɿʃɼɸʋʀʔʌɲɶʅɹʆɻʍʐʆɳʌʏɻʍɻͿȰʆʉɿʍʐʆɲʌʏɼʍɸɿʎ f1 f2 ɸʀʆɲɿʉʌɿʍʅɹʆɸʎʍʚɹʆɲʍʑʆʉʄʉȴʇɸʀʆɲɿ
ɹʆɲɸʍʘʏɸʌɿʃʊʍɻʅɸʀʉɼɳʃʌʉʏʉʐȴɻf1 ɸʀʆɲɿʔʌɲɶʅɹʆɻʍɸʅɿɲʋɸʌɿʉʖɼʏʉʐ ʇ ʃɲɿ lim f x 0
x2 =
"p^ h ʏʊʏɸ ɿʍʖʑɸɿ ʊʏɿ lim f x f x 0
x1 2 =
"p^ ^^ h hh ;ȴɻʄɲɷɼ ʏʉ
ɶɿʆʊʅɸʆʉʅɻɷɸʆɿʃɼʎʍʐʆɳʌʏɻʍɻʎɸʋʀʔʌɲɶʅɹʆɻɸʀʆɲɿʅɻɷɸʆɿʃɼͿɀɸɽʉɷʉʄʉɶʀɸʎ
ɉʋʉʄʉɶɿʍʅʊʎʉʌʀʘʆʍʏʉx0 :ŝ Ȱʋʄɳʊʌɿɲ
ɇɸɲʐʏɳɷɸʆʐʋɳʌʖɸɿʋɲʌʉʆʉʅɲʍʏɼʎɸʀʏɸɲʐʏʊʎɷɸʆʅɻɷɸʆʀɺɸʏɲɿȳɿɲʆɲʏɲʐʋʉʄʉɶʀʍʉʐʅɸɽɹʏʉʐʅɸɲʋɸʐɽɸʀɲʎʊʋʉʐdžʏʉx0 ʃɲɿɸʃʏɸ-ʄʉʑʅɸʏɿʎʋʌɳʇɸɿʎ
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim xx
13 2
x 2 ++
".
ȳɿɲʆɲʐʋʉʄʉɶʀʍʉʐʅɸʏʉʊʌɿʉɽɲɲʆʏɿʃɲʏɲʍʏʀʍʉʐʅɸʏʉdžʅɸʏʉϮʃɲɿɽɲʃɳʆʉʐʅɸʋʌɳʇɸɿʎ lim x
x1
3 22 1
3 2 238
x 2
$++ = +
+ ="
.
ŝŝ Ʉʌɿɲ 00 ʅɸʋʉʄʐʙʆʐʅɲ
ɇɸ ɲʐʏɳ ɽɲ ʋɲʌɲɶʉʆʏʉʋʉɿʉʑʅɸ ɲʌɿɽʅɻʏɼ ʃɲɿ ʋɲʌʉʆʉʅɲʍʏɼ ʅɸʏɻɴʉɼɽɸɿɲʏʉʐʍʖɼʅɲʏʉʎ,ŽƌŶĞƌ;ʍʏʉ x0Ϳɼɳʄʄɻʎʅɸɽʊɷʉʐʋɲ-ʌɲɶʉʆʏʉʋʉʀɻʍɻʎʅɹʖʌɿʎʊʏʉʐʆɲɲʋʄʉʋʉɿɻɽɸʀʉʊʌʉʎ x x0- ʋʉʐʋʌʉʃɲʄɸʀʏɻʆɲʋʌʉʍɷɿʉʌɿʍʏʀɲȾɲʏʊʋɿʆɽɹʏʉʐʅɸʊʋʉʐdžʏʉx0 ʃɲɿʃɳʆʉʐʅɸʋʌɳʇɸɿʎ
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x xx x x
3 104 5 2
x 2 2
3 2
+ -- + -
".
ɅɲʌɲʏɻʌʉʑʅɸʊʏɿɲʆɲʆʏɿʃɲʏɲʍʏɼʍʉʐʅɸʊʋʉʐdžʏʉϮʋʌʉʃʑʋʏɸɿ 00
5
ɲʋʌʉʍɷɿʉʌɿʍʏʀɲȾɳʆʉʐʅɸʍʖɼʅɲ,ŽƌŶĞƌɶɿɲʆɲʋɲʌɲɶʉʆʏʉʋʉɿɼ-ʍʉʐʅɸʏʉʆɲʌɿɽʅɻʏɼʃɲɿɴʌʀʍʃʉʐʅɸʏɿʎʌʀɺɸʎʏʉʐʋɲʌʉʆʉʅɲʍʏɼɶɿɲʆɲʏʉʆʋɲʌɲɶʉʆʏʉʋʉɿɼʍʉʐʅɸʉʋʊʏɸʋɲʀʌʆʉʐʅɸ
lim limx xx x x
x xx x x
3 104 5 2
2 52 2 1
x x2 2
3 2
2
2
+ -- + - = - +
- - +=
" " ] ]] ] g gg glim x
x x5
2 12 5
2 2 2 171
x 2
2 2 $+
- + = +- + =
"
ŝŝŝɄʌɿɲ 00 ʅɸʌʀɺɸʎ
ȺɲʋʉʄʄɲʋʄɲʍɿɳɺʉʐʅɸɲʌɿɽʅɻʏɼʃɲɿʋɲʌʉʆʉʅɲʍʏɼʅɸʏɻʍʐɺʐɶɼʋɲʌɳʍʏɲʍɻɅɲʌɳʍʏɲʍɻ ɇʐɺʐɶɼʎ Ȱʋʉʏɹʄɸʍʅɲ
x a- x a+ x 2a-
x3 a- x x23 3 2a a+ + x 3a-x a- x x xv v v v1 2 2 1fa a a+ + + +- - - - xv va-
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim xx 100 10
x 0 2
2 + -"
.Ʌɲʌɲʏɻʌʉʑʅɸʊʏɿɲʆɲʆʏɿʃɲʏɲʍʏɼʍʉʐʅɸʊʋʉʐ dž ʏʉϬʋʌʉʃʑʋʏɸɿ
00 ɲʋʌʉʍɷɿʉʌɿʍʏʀɲȺɲʋʌɹʋɸɿʆɲʋʉʄʄɲʋʄɲʍɿɳʍʉʐʅɸɲʌɿɽʅɻʏɼʃɲɿʋɲʌʉʆʉʅɲʍʏɼʅɸʏɻʍʐɺɻɶɼʋɲʌɳʍʏɲʍɻʏʉʐɲʌɿɽʅɻʏɼʉʋʊʏɸʋʌʉʃʑʋʏɸɿ
lim limxx
xx
xx100 10 100 10
100 10100 10
x x0 2
2
0 2
2
2
2
$+ - = + -
+ ++ + =
" "e o
lim limx xx
x xx
100 10100 100
100 10x x0 2 2
2
0 2 2
2
+ ++ - =
+ +=
" "^ ^h hlim lim
x xx
x100 10 100 101
0 1001
201
10x x0 2 2
2
0 2+ +=
+ +=
+=
+" "^ hʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim
xx
232
3x 6 -
--
+"
.
Ʌɲʌɲʏɻʌʉʑʅɸʊʏɿɲʆɲʆʏɿʃɲʏɲʍʏɼʍʉʐʅɸʊʋʉʐ dž ʏʉϲʋʌʉʃʑʋʏɸɿ
00 ɲʋʌʉʍɷɿʉʌɿʍʏʀɲȺɲʋʌɹʋɸɿʆɲʋʉʄʄɲʋʄɲʍɿɳʍʉʐʅɸɲʌɿɽʅɻʏɼʃɲɿʋɲʌʉʆʉʅɲʍʏɼʅɸʏɻʍʐɺɻɶɼʋɲʌɳʍʏɲʍɻʏʉʐɲʌɿɽʅɻʏɼʃɲɿʏʉʐʋɲʌʉʆʉʅɲʍʏɼʉʋʊʏɸʋʌʉʃʑʋʏɸɿ
lim limxx
xx
xx
xx
2 23 3
2 23 3
33
22
33
22
x x6 6$ $
- -+ - =
- -+ -
++
--
++
++ =
" "e o
6
lim limxx
xx
xx
xx
2 43 9
3 32 2
66
3 32 2
x x6 6$ $- -
+ -+ +- + = -
-+ +- + =
" "e eo o
lim limxx
xx
xx
66
3 32 2
3 32 2
6 3 36 2 2
3 34 2 1
x x6 6$-
-+ +- + =
+ +- + =
+ +- + = +
+ =" "e o .
Ȱʆʃɳʋʉɿʉʎʊʌʉʎʋɸʌɿɹʖɸɿʋɸʌɿʍʍʊʏɸʌɲɲʋʊɷʑʉʌɿɺɿʃɳʏʊʏɸɶɿɲʏɻʆɸʑʌɸʍɻʏʉʐʉʌʀʉʐɲʔɲɿʌʉʑʅɸɲʋʊʃɳɽɸʌɿɺɿʃʊʏɻʆ ʏɿʅɼʏʉʐɶɿɲx x0= ȶʋɸɿʏɲɸʀʏɸɷɿɲʍʋɳʅɸʏʉʃʄɳʍʅɲʃɲɿʋʉʄʄɲʋʄɲʍɿɳɺʉʐʅɸʏʉʃɲɽɹʆɲʅɸʏɻʍʐɺʐɶɼʋɲʌɳʍʏɲʍɻɸʀʏɸɷɿɲɿʌʉʑʅɸʊʄʉʐʎʏʉʐʎʊʌʉʐʎʋʉʐɷɻʅɿʉʐʌɶʉʑʆʏɲɿʅɸx x0- .
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim xx x
22 7 1
x 2 -+ - + +
".
ɅɲʌɲʏɻʌʉʑʅɸʊʏɿɲʆɲʆʏɿʃɲʏɲʍʏɼʍʉʐʅɸʊʋʉʐdžʏʉϮʋʌʉʃʑʋʏɸɿ 00
ɲʋʌʉʍɷɿʉʌɿʍʏʀɲɃʋʊʏɸɽɲʋʌɹʋɸɿʆɲɷɿɲʍʋɳʍʉʐʅɸʏʉʃʄɳʍʅɲʍɸɷʑʉʃɲɿʆɲʋʉʄʄɲʋʄɲʍɿɳʍʉʐʅɸʏʉʃɲɽɹʆɲʅɸʏɻʍʐɺʐɶɼʋɲʌɳʍʏɲ-ʍɻʉʋʊʏɸʋʌʉʃʑʋʏɸɿ
lim xx
xx
22 2
27 3
x 2 -+ - - -
+ - ="c m
limx xx x
x xx x
2 2 22 2 2 2
2 7 37 3 7 3
x 2 - + ++ - + + -
- + ++ - + + =
" ^ ^^ ^
^ ^^ ^e h h
h hh hh h o
limx x
xx x
x2 2 2
22 7 3
2x 2 - + +
--
- + +-
=" ^ ^ ^ ^e h h h h olim
x x2 21
7 31
41
61
121
x 2
+ +-
+ += - =
"c m .
ŝǀ Ʉʌɿɲ 00 ʅɸɲʋʊʄʐʏɲ
Ȱʆʏʉx0 ɷɸʆʅɻɷɸʆʀɺɸɿʏʉɲʋʊʄʐʏʉʏʊʏɸɴʌʀʍʃʉʐʅɸʏʉʋʌʊʍɻʅʉʏɻʎʋɲʌɳʍʏɲʍɻʎʅɹʍɲʍɸɲʐʏʊɶɿɲx x0= ʃɲɿɷɿʙʖʆʉʐʅɸʏʉɲʋʊʄʐʏʉ
x2
3 1x 2
- -
Ʌɲʌɲʏɻʌʉʑʅɸʊʏɿɲʆɲʆʏɿʃɲʏɲʍʏɼʍʉʐʅɸʊʋʉʐ dž ʏʉϮʋʌʉʃʑʋʏɸɿ
00 ɲʋʌʉʍɷɿʉʌɿʍʏʀɲɉʋʉʄʉɶʀɺʉʐʅɸʏʉlim(x - 3) = -1 < 0.
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x - .
ȵʋʉʅɹʆʘʎʇɲʆɲɶʌɳʔʉʐʅɸʏʉɲʌʖɿʃʊʊʌɿʉɷɿʙʖʆʉʆʏɲʎʏʉɲʋʊʄʐʏʉʃɲɿɲʄʄɳ-ɺʉʆʏɲʎʏɲʋʌʊʍɻʅɲʏɻʎʋɲʌɳʍʏɲʍɻʎʅɹʍɲʍʏʉɲʋʊʄʐʏʉ;ɸʋɸɿɷɼɻʏɿʅɼʏʉʐʉʌʀʉʐɸʀʆɲɿɲʌʆɻʏɿʃɼͿʉʋʊʏɸʋʌʉʃʑʋʏɸɿ
lim lim lim xx
xx
xx
22
23 1
22 1
x x x2 2 2 --
-- - -
= -- + =
-=-
" " "
^ ^h h.
Ȱʆʏʉx0 ʅɻɷɸʆʀɺɸɿʏʉɲʋʊʄʐʏʉʏʊʏɸʋɲʀʌʆʉʐʅɸʋʄɸʐʌɿʃɳʊʌɿɲȴɸʆ
"
"
x 2
7
ʇɸʖʆɳʅɸʊʏɿʊʏɲʆʏʉx x0" + ʏʉx x> 0 ʃɲɿʊʏɲʆʏʉx x0" - ʏʉx x< 0 .
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim xx x x3 2
11
x 1
2
-- + - +
".
Ʌɲʌɲʏɻʌʉʑʅɸʊʏɿɲʆɲʆʏɿʃɲʏɲʍʏɼʍʉʐʅɸʊʋʉʐ dž ʏʉϭʋʌʉʃʑʋʏɸɿ
00 ɲʋʌʉʍɷɿʉʌɿʍʏʀɲ ɉʋʉʄʉɶʀɺʉʐʅɸ ʏʉ lim x x3 2 0
x
2
1- + =
" ȵʋʉʅɹ-
ʆʘʎʋʌɹʋɸɿʆɲʋɳʌʉʐʅɸʋʄɸʐʌɿʃɳʊʌɿɲɉʋʉʄʉɶʀɺʉʐʅɸʏɿʎʌʀɺɸʎʏɻʎx x3 2 02 - + = ʃɲɿʃɳʆʉʐʅɸʋʀʆɲʃɲʋʌʉʍɼʅʘʆ
x -∞ 1 2 +∞+ - +
Ʌɲʌɲʏɻʌʉʑʅɸʊʏɿ x x3 2 0<2 - + ʊʏɲʆʏʉ x 1" + ʃɲɿ x x3 2 0>2 - + ʊʏɲʆʏʉx 1" - ɉʋʉʄʉɶʀɺʉʐʅɸʏʙʌɲʏɲʋʄɸʐʌɿʃɳʊʌɿɲʉʋʊʏɸʋʌʉ-ʃʑʋʏɸɿ
lim limxx x x
xx x x
13 2 1
11 2 1
x x1
2
1-- - + - + = -
- - - - - =" "+ +
^ ^ ^ ^h h h hlim x
x x1
11 1
2 10
x 1 -- -
=- -- +
=" +
^ ^ ^h h hlim limx
x x xx
x x x1
3 2 11
1 2 1x x1
2
1-- + - + = -
- - - - =" "- +
^ ^ ^h h hlim x
x x1
1 2 11 3 2
x 1 -- -
= - =--
" +
^ ^h hȵʋɸɿɷɼ ʏɲ ʋʄɸʐʌɿʃɳ ʊʌɿɲ ɸʀʆɲɿ ɷɿɲʔʉʌɸʏɿʃɳ ʅɸʏɲʇʑ ʏʉʐʎ ʏʉ
lim xx x x
13 2 1
x 1
2
-- + - +
"ɷɸʆʐʋɳʌʖɸɿ
ǀ Ʉʌɿɲ 00 ʅɸʏʌɿɶʘʆʉʅɸʏʌɿʃʉʑʎʊʌʉʐʎ
Ⱥɲ ʋʌʉʍʋɲɽʉʑʅɸ ;ʋʉʄʄɲʋʄɲʍɿɳɺʉʆʏɲʎ ʃɲɿ ɷɿɲɿʌʙʆʏɲʎͿ ʆɲ ɸʅ-
ʔɲʆʀɺʉʐʅɸʊʌɿɲ ʏɻʎʅʉʌʔɼʎ lim f xf x
x x0
hn" ] ]g g ɼ lim f x
f x 1x x0
vyo -" ]] gg ʊʋʉʐ
lim f x 0x x0
="] g ȶʏʍɿɽɹʏʉʆʏɲʎ f x u=] g ɽɲʃɲʏɲʄɼɶʉʐʅɸʍʏɲɴɲʍɿʃɳ
ʊʌɿɲ lim uu 1
u 0
hn ="
ɼ lim uu 1 0
u 0
vyo - ="
.
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim xx3
x 0
hn"
.
lim limxx
xx3 3 3
3x x0 0
hn hn=" "
: DȺɹʏʉʐʅɸu x3= .Ʉʏɲʆx 0" ʏʉu 0" .
ȵʋʉʅɹʆʘʎ lim uu3 3 1 3
u 0$
hn = =": D .
x2 - 3x + 2
8
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim xxππ
x π
hn--
"
] g.
Ⱥɹʏʉʐʅɸu x π= - .Ʉʏɲʆx π" ʏʉu 0" .ȵʋʉʅɹʆʘʎ lim u
u 1u 0
hn ="
.ǀŝɄʌɿɲʅɸʍʐʆɲʌʏɼʍɸɿʎʋʉʄʄɲʋʄʉʑʏʑʋʉʐʍʏʉʍɻʅɸʀʉɲʄʄɲɶɼʎʏʑ-
ʋʉʐʏɻʎʍʐʆɳʌʏɻʍɻʎȺɲʐʋʉʄʉɶʀɺʉʐʅɸʏɲʋʄɸʐʌɿʃɳʊʌɿɲʏɻʎĨʍʏʉx0 ȶʏʍɿɶɿɲʆɲʐʋɳʌ-ʖɸɿʏʉʊʌɿʉɽɲʋʌɹʋɸɿʏɲʋʄɸʐʌɿʃɳʊʌɿɲʆɲɸʀʆɲɿʀʍɲ
ʋʖ ȳɿɲʋʉɿɲʏɿʅɼʏʉʐ R!a ʐʋɳʌʖɸɿʏʉʊʌɿʉʏɻʎ,,f x
x xx x1 3
2 3<2
Ha=
-^ h ( ʍʏʉʍɻʅɸʀʉx 3= ;ȳɿɲʆɲʐʋɳʌʖɸɿʏʉʊʌɿʉʍʏʉx 3= ɽɲʋʌɹʋɸɿʆɲʐʋɳʌʖʉʐʆʏɲʋʄɸʐ-ʌɿʃɳʊʌɿɲʃɲɿʆɲɸʀʆɲɿʀʍɲlim lim xf x 1 3 1 8
xx 3
2
3
2-= = - ="" --
^ ^h hlim limf x x2 2 3 6x x3 3
a a a= = =" "+ +^ ^h h
Ʌʌɹʋɸɿ6 8 34+a a= = .
ǀŝŝȰʆʏɿʃɲʏɳʍʏɲʍɻȰʆɽɹʄʉʐʅɸʆɲʐʋʉʄʉɶʀʍʉʐʅɸɹʆɲʊʌɿʉʏɻʎʅʉʌʔɼʎ limA f g x
x x0=
"]^ gh
ʏʊʏɸ ɽɹʏʉʐʅɸ g x u=] g ʉʋʊʏɸ limg x ux x 0
0=
"] g ʃɲɿ ɲʆu u0! ʃʉʆʏɳ
ʍʏʉ x0 ʏʊʏɸ limA f uu u0
="] g ȵʋʉʅɹʆʘʎ ɲʆʏʀ ʆɲ ʐʋʉʄʉɶʀʍʉʐʅɸ ʏʉ
lim f g xx x0"
]^ ghʐʋʉʄʉɶʀɺʉʐʅɸʏʉɸʐʃʉʄʊʏɸʌʉʊʌɿʉ lim f uu u0"] g.
ʋʖ Ȱʆ lim f x 1x 0
="] g ʆɲʐʋʉʄʉɶɿʍʏɸʀʏʉ lim f x 2
x 2-
"] g.
Ⱥɹʏʉʐʅɸu x 2= - .Ʉʏɲʆx 2" ʏʉu 0" .ȵʋʉʅɹʆʘʎ lim f u 1
u 0=
"] g .
3.3 ȺɸʘʌɻʏɿʃɳȺɹʅɲʏɲʅɸʊʌɿɲɄʏɲʆʅɲʎɺɻʏʉʑʆʏʉʊʌɿʉʅɿɲʎʍʐʆɳʌʏɻʍɻʎʏɻʎʉʋʉʀɲʎɷɸʆɶʆʘʌʀ-
ɺʉʐʅɸʏʉʆʏʑʋʉɽɲɸʔɲʌʅʊɺʉʐʅɸʃʌɿʏɼʌɿʉʋɲʌɸʅɴʉʄɼʎɲʆɹʖʉʐʅɸɷɿʋʄɼɲʆɿʍʊʏɻʏɲ
ȰʆɹʖʉʐʅɸʊʅʘʎʏʉʊʌɿʉʅɿɲʎʋɲʌɳʍʏɲʍɻʎɻʉʋʉʀɲɽɲʋɸʌɿɹʖɸɿʏɻʆĨ ʃɲɿʅɲʎɺɻʏʉʑʆʏʉʊʌɿʉʏɻʎĨ ɽɲɽɹʏʉʐʅɸʏɻʆʋɲʌɳʍʏɲʍɻh x] gʃɲɿɽɲʄʑʆʉʐʅɸʘʎʋʌʉʎf x] gȶʋɸɿʏɲɽɲʐʋʉʄʉɶʀɺʉʐʅɸʏʉʊʌɿʉʏɻʎf x^ hɲʔʉʑʊʄɲʏɲʐʋʊʄʉɿʋɲʊʌɿɲɸʀʆɲɿɶʆʘʍʏɳ
3.4 ɀɻɅɸʋɸʌɲʍʅɹʆʉɄʌɿʉʍʏʉx R0 !
x x0"
ȶʍʏʘʍʐʆɳʌʏɻʍɻ :f A " R Ȼʍʖʑʉʐʆʉɿɸʇήʎɿɷɿʊʏɻʏɸʎ ͻ Ȱʆlim f]xg =+3ʏʊʏɸf]xg > 0ʃʉʆʏɳʍʏʉx0 .
9
ͻ Ȱʆ lim f xx x0
3=-"] g ʏʊʏɸf x 0<] g ʃʉʆʏɳʍʏʉx0 .
ͻ Ȱʆ lim f xx x0
3=+"] g ʏʊʏɸ lim f x
x x03- =-
"]^ gh .
ͻ Ȱʆ lim f xx x0
3=-"] g ʏʊʏɸ lim f x
x x03- =+
"]^ gh .
ͻ Ȱʆ lim f x 0x x0
="] g ʃɲɿf x 0>] g ʃʉʆʏɳʍʏʉx0 ʏʊʏɸ lim f x
1x x0
3=+" ] g .
ͻ Ȱʆ lim f x 0x x0
="] g ʃɲɿf x 0<] g ʃʉʆʏɳʍʏʉx0 ʏʊʏɸ lim f x
1x x0
3=-" ] g .
ͻ Ȱʆ lim f xx x0
3=+"] g ɼ 3- ʏʊʏɸ lim f x
1 0x x0
=" ] g .
ͻ Ȱʆ lim f xx x0
3=+"] g ʏʊʏɸ lim f x
x x03=+
"
l ] g .ͻ Ȱʆf x g xG] ]g gʃʉʆʏɳʍʏʉx0 ʃɲɿ lim f x
x x03=+
"] g ʏʊʏɸ limg x
x x03=+
"] g .
ͻ Ȱʆf x g xH] ]g gʃʉʆʏɳʍʏʉx0 ʃɲɿ lim f xx x0
3=-"] g ʏʊʏɸ limg x
x x03=-
"] g .
Ʌɲʌɲʏɻʌɼʍɸɿʎͻ ȳɿɲʄʊɶʉʐʎʍʐʆʏʉʅʀɲʎʅʋʉʌʉʑʅɸɲʆʏʀʏɻʎɹʃʔʌɲʍɻʎΗ lim f]xg = 0
x x0=
"
ʃɲɿf]xg > 0ʃʉʆʏɳʍʏʉx0 Ηʆɲʖʌɻʍɿʅʉʋʉɿɼʍʉʐʅɸʏʉʍʐʅɴʉʄɿʍʅʊ 0+ Ȱʆʏʀʍʏʉɿʖɲ ʅʋʉʌʉʑʅɸ ɲʆʏʀ ʏɻʎ ɹʃʔʌɲʍɻʎ Ηlim f]xg 0 ʃɲɿf]xg < 0ʃʉʆʏɳʍʏʉx0 Ηʆɲʖʌɻʍɿʅʉʋʉɿɼʍʉʐʅɸʏʉʍʐʅɴʉʄɿʍʅʊ0-.
ɀʉʌʔɹʎʉʌʀʘʆ
lim f xx x0"] g 0>a 0<a 0>a 0<a 0 0 3+ 3+ 3- 3-
limg xx x0"] g 3+ 3+ 3- 3- 3+ 3- 3+ 3- 3+ 3-
lim f x g xx x0
+"] ]^ g gh 3+ 3+ 3- 3- 3+ 3- 3+ ? ? 3-
lim f x g xx x0
$"] ]^ g gh 3+ 3- 3- 3+ ? ? 3+ 3- 3- 3+
Ȱʋʌʉʍɷɿʊʌɿʍʏɸʎɀʉʌʔɹʎ
3 3+ + -] ]g g 3 3+ - +] ]g g 3 3- - -] ]g g0 $ !3] g 00 !!33
00 !3 !3^ h ϭ!3 0!3^ hɀɸɽʉɷʉʄʉɶʀɸʎ
Ɉɲʅɻʋɸʋɸʌɲʍʅɹʆɲʊʌɿɲʍʏʉx R0 ! ɽɲɸʀʆɲɿɶɸʆɿʃɳʊʌɿɲʏɻʎʅʉʌ-
ʔɼʎlim g xf x
x x0" ]] gg ʅɸf x 00 !a=] g ʃɲɿg x 00 =] g ȳɿɲʆɲʏɲʐʋʉʄʉɶʀʍʉʐ-
ʅɸɽɲɲʋʉʅʉʆʙʆʉʐʅɸʏʉʆʋɲʌʉʆʉʅɲʍʏɼɽɲʐʋʉʄʉɶʀɺʉʐʅɸʏʉʊʌɿʉʏʉʐʃɲɿɽɲɸʇɸʏɳɺʉʐʅɸʏʉʋʌʊʍɻʅʊʏʉʐ
ŝ Ȱʆʏʉʋʌʊʍɻʅʉɸʀʆɲɿʍʏɲɽɸʌʊʃʉʆʏɳʍʏʉx0 ʏʊʏɸɲʆɳʄʉɶɲʅɸʏʉʆʋʀʆɲʃɲʍʏɿʎʅʉʌʔɹʎʉʌʀʘʆɴʌʀʍʃʉʐʅɸɲʋɸʐɽɸʀɲʎʏʉʊʌɿʉ
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀɲʆʐʋɳʌʖɸɿʏʉʊʌɿʉ lim x 21
x 2 2-" ^ h .ɈʉʊʌɿʉʏʉʐʋɲʌʉʆʉʅɲʍʏɼʊʏɲʆʏʉdžʏɸʀʆɸɿʍʏʉϮɸʀʆɲɿʅɻɷɹʆɃ
"x x0
10
ʋɲʌʉʆʉʅɲʍʏɼʎ ʃʉʆʏɳʍʏʉ Ϯ ɸʀʆɲɿ ʋɳʆʏʉʏɸɽɸʏɿʃʊʎ ;ɸʋɸɿɷɼ ɸʀʆɲɿʐʗʘʅɹʆʉʎʍʏʉʏɸʏʌɳɶʘʆʉͿȵʋʉʅɹʆʘʎʏʉʊʌɿʉɽɲɿʍʉʑʏɸʅɸ 3+ .Ʌʉɿʉʍʑʆʏʉʅɲʅʋʉʌʉʑʅɸʆɲɶʌɳʗʉʐʅɸ lim x 2
101
x 2 2 3- = =+"
+^ h .
ŝŝ Ȱʆ ʉ ʋɲʌʉʆʉʅɲʍʏɼʎ ɲʄʄɳɺɸɿ ʋʌʊʍɻʅʉ ɸʃɲʏɹʌʘɽɸʆ ʏʉʐ x0 ʏʊʏɸʋɲʀʌʆʉʆʏɲʎ ʋʄɸʐʌɿʃɳ ʊʌɿɲ ɲʃʉʄʉʐɽʉʑʅɸ ʏɻʆ ʋʌʉɻɶʉʑʅɸʆɻ ʅɸ-ɽʉɷʉʄʉɶʀɲʃɲɿʏɸʄɿʃɳɴʌʀʍʃʉʐʅɸʊʏɿʏʉʊʌɿʉɷɸʆʐʋɳʌʖɸɿɸʋɸɿɷɼʏɲʋʄɸʐʌɿʃɳʊʌɿɲɸʀʆɲɿɷɿɲʔʉʌɸʏɿʃɳ
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀɲʆʐʋɳʌʖɸɿʏʉʊʌɿʉ lim x 21
x 2 -".
ɈʉʊʌɿʉʏʉʐʋɲʌʉʆʉʅɲʍʏɼʊʏɲʆʏʉdžʏɸʀʆɸɿʍʏʉϮɸʀʆɲɿʅɻɷɹʆȵʋɸɿ-ɷɼɷɸʆʅʋʉʌʉʑʅɸʆɲʋʌʉʍɷɿʉʌʀʍʉʐʅɸʏʉʋʌʊʍɻʅʉʏʉʐʋɲʌʉʆʉʅɲ-ʍʏɼʃʉʆʏɳʍʏʉϮɽɲʋɳʌʉʐʅɸʋʄɸʐʌɿʃɳʊʌɿɲlim x 2
101
x 23- = =+
"+
+ lim x 2
101
x 23- = =-
"-
-
Ɉɲʋʄɸʐʌɿʃɳʊʌɿɲ ɸʀʆɲɿ ɷɿɲʔʉʌɸʏɿʃɳʅɸʏɲʇʑ ʏʉʐʎ ɸʋʉʅɹʆʘʎ ʏʉʊʌɿʉ lim x 2
1x 2 -"
ɷɸʆʐʋɳʌʖɸɿ
3.5 Ʉʌɿɲʍʏʉȱʋɸɿʌʉͻ ȲɲʍɿʃɳɄʌɿɲ
ͻ limxx
v 3=+" 3+
.
ͻ,, ,limxvv
22 1 Z
xv 3
3!
ll
l=+ =- = +" 3-( .
ͻ lim x1 0
xv =
"!3.
ͻ Ʉʌɿʉʋʉʄʐʘʆʐʅɿʃɼʎͻ lim limx x x x
xv
vv
v
xv
v1
11 0fa a a a a+ + + + =
" "! !3 3-
-] ]g g,ʅɸ 0v !a .ͻ Ʉʌɿʉʌɻʏɼʎ
ͻ
,,
,lim limx x x
x x xxx
0
>
<x
vv
vv
x
vv
11
1 0
11
1 0
!
ff
3
b b b ba a a a
ba
o l
bao l
o l+ + + ++ + + + = = =
" "! !3 3ll
ll
ll
l
o
--
--
Z
[
\
]]
], ʅɸ
0v !a bl .ͻ Ʉʌɿʉɳʌʌɻʏɻʎ
ͻ lim x x xx v
vv
v1
11 0fa a a a+ + + + =
"!3
l-
-
lim x x x xxv
vv
v v1
11 0fa a a a+ + + + =
"!3
l --b l
lim x x x xx
vv
vv v
111 0$ fa a a a+ + + + =
"!3
ll
--b l
0 0 0v$3 f 3a+ + + + + =+l] g .
11
ͻ Ʉʌɿʉɸʃɽɸʏɿʃɼʎͻ Ȱʆ 1>a ʏʊʏɸ lim
xx 3a =+
" 3+ʃɲɿ lim 0
xxa =
" 3-.
ͻ Ȱʆ0 1< <a ʏʊʏɸ lim 0x
xa =" 3+
ʃɲɿ limx
x 3a =+" 3-
.ͻ Ʉʌɿʉʄʉɶɲʌɿɽʅɿʃɼʎ
ͻ Ȱʆ 1>a ɲʏʊʏɸ limlog xx
3=+" 3 a+
ʃɲɿ limlog xx 0
3=-"
a+.
ͻ Ȱʆ0 1< <a ʏʊʏɸ limlog xx
3=-" 3 a+
ʃɲɿ limlog xx 0
3=+"
a+.
ɀɸɽʉɷʉʄʉɶʀɸʎ
ŝ ɇʏɲʊʌɿɲʍʏʉɳʋɸɿʌʉʊʏɲʆɹʖʉʐʅɸɲʋʌʉʍɷɿʉʌɿʍʏʀɲ !!33 ɶɸʆɿʃɳ
ɴɶɳɺʉʐʅɸʃʉɿʆʊʋɲʌɳɶʉʆʏɲʋʌʉʍʋɲɽʙʆʏɲʎʆɲɷɻʅɿʉʐʌɶɼʍʉʐʅɸʅɻɷɸʆɿʃɳʃɲɿʆɲʅʋʉʌɹʍʉʐʅɸʆɲɷɿʙʇʉʐʅɸʏɻʆɲʋʌʉʍɷɿʉʌɿʍʏʀɲȰʆɹʖʉʐʅɸʋʉʄʐʙʆʐʅɲʅʋʉʌʉʑʅɸʆɲʖʌɻʍɿʅʉʋʉɿɼʍʉʐʅɸɲʋɸʐɽɸʀɲʎʏɿʎɿɷɿʊʏɻʏɸʎ
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x x3 5 2x
2 + -" 3+^ h.
ȶʖʉʐʅɸlim limx x x3 5 2 3x x
2 2 3+ - = =+" "3 3+ +^ ^h h .
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim xx x
13 5 2
x
2
-+ -
" 3-.
ȶʖʉʐʅɸ
lim lim limxx x
xx x1
3 5 2 3 3x x x
2 2
3-+ - = = =-
" " "3 3 3- - -^ h .
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x xx x6 13 5 2
x 2
2
- ++ -
" 3-.
ȶʖʉʐʅɸ
lim limx xx x
xx
6 13 5 2
63
63
21
x x2
2
2
2
- ++ - = = =
" "3 3- -.
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x xx
3 5 21
x 2 - +-
" 3+.
ȶʖʉʐʅɸlim lim limx x
xxx
x3 5 21
3 31 0
x x x2 2- +- = = =
" " "3 3 3+ + +.
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x x5 2x
2 + -" 3+
.ȶʖʉʐʅɸ
lim limx x x x x5 2 1 5 2x x
2 22+ - = + - =
" "3 3+ +b l lim x x x1 5 2
x 2+ - =" 3+b l
lim x x x1 5 2 1x 2 $3 3+ - =+ =+" 3+b l .
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x x5 2x
2 + -" 3-
.ȶʖʉʐʅɸ
lim limx x x x x5 2 1 5 2x x
2 22+ - = + - =
" "3 3- -b l lim x x x1 5 2
x 2+ - =" 3-b l
12
lim x x x1 5 2 1x 2 $3 3- + - =- - =+" 3-b ]l g .
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x x x5 2x
2 + - +" 3+^ h.
ȶʖʉʐʅɸ
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 x1 5 2 1 5 2x x2 2+ - + = + - + =" "3 3+ +
bc bcl m l mlim x x x1 5 2 1 2x 2 $3 3+ - + =+ =+" 3+
bc l m; E .
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim x x x5 2x
2 + - +" 3-^ h.
Ȱʆɴɶɳʄʉʐʅɸʃʉɿʆʊʋɲʌɳɶʉʆʏɲʏʉdžʊʋʘʎʃɳʆɲʅɸʍʏʉʋʌʉɻɶʉʑ-ʅɸʆʉʋɲʌɳɷɸɿɶʅɲɽɲʋʌʉʃʑʗɸɿɲʋʌʉʍɷɿʉʌɿʍʏʀɲ0 $3ȵʋʉʅɹʆʘʎʋʉʄʄɲʋʄɲʍɿɳɺʉʐʅɸʋʌʙʏɲʅɸʏɻʍʐɺʐɶɼʋɲʌɳʍʏɲʍɻʏʉʐɲʌɿɽʅɻʏɼʃɲɿɹʖʉʐʅɸ
lim limx x xx x xx x x5 2
5 25 2
x x2
2
2 2
+ - + =+ - -+ - =-
" "3 3- -^ h
lim limx x x x
x
x x x x
x x
1 5 25 2
1 5 25 2
x x
2 2+ - -=
- + - -=- -
" "3 3- -bc bb
l m ll
limx x x
x x
1 5 2 1
5 2
1 0 0 15 0
25
x
2- + - +=- + - +
=-- -
" 3+ bcb
^l ml
h .
ŝŝ ɇʏɲʊʌɿɲʅɸɸʃɽɸʏɿʃɹʎɲʆɹʖʉʐʅɸɸʃɽɸʏɿʃɹʎʅɸʏɻʆʀɷɿɲɴɳʍɻʏʊʏɸͻ Ȱʆʏʉx " 3+ ʃɲɿɻɴɳʍɻɸʀʆɲɿʅɸɶɲʄʑʏɸʌɻʏʉʐϭʏʊʏɸɽɲɴɶɳ-
ɺʉʐʅɸʃʉɿʆʊʋɲʌɳɶʉʆʏɲʏɻʆɸʃɽɸʏɿʃɼͻ Ȱʆʏʉx " 3- ʃɲɿɻɴɳʍɻɸʀʆɲɿʅɸɶɲʄʑʏɸʌɻʏʉʐϭʏʊʏɸɲʆʏɿʃɲɽɿ-
ʍʏʉʑʅɸɲʋɸʐɽɸʀɲʎɲʔʉʑʏʉʊʌɿʉʏɻʎɸʃɽɸʏɿʃɼʎʃɳʆɸɿϬͻ Ȱʆʏʉx " 3+ ʃɲɿɻɴɳʍɻɸʀʆɲɿʅɿʃʌʊʏɸʌɻʏʉʐϭʏʊʏɸɲʆʏɿʃɲɽɿ-
ʍʏʉʑʅɸɲʋɸʐɽɸʀɲʎɲʔʉʑʏʉʊʌɿʉʏɻʎɸʃɽɸʏɿʃɼʎʃɳʆɸɿϬͻ Ȱʆʏʉx " 3- ʃɲɿɻɴɳʍɻɸʀʆɲɿʅɿʃʌʊʏɸʌɻʏʉʐϭʏʊʏɸɽɲɴɶɳɺʉʐ-
ʅɸʃʉɿʆʊʋɲʌɳɶʉʆʏɲʏɻʆɸʃɽɸʏɿʃɼʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim e
e2 1
3x x
x
++
" 3+.
ȶʖʉʐʅɸ
lim lim limee
e e
e e
e
e2 1
32 11 3
2 11 3
21
x x
x
x xx
xx
xx
x
++ =
+
+=
+
+=
" " "3 3 3+ + +``
jj
.
13
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim ee2 2
3x x
x
++
" 3-.
ȶʖʉʐʅɸlim e
e2 2
32 00 3
23
x x
x
++ = +
+ =" 3-
.ŝŝŝɇʏɲʊʌɿɲʅɸɸʃɽɸʏɿʃɹʎʅɸɷɿɲʔʉʌɸʏɿʃɹʎɴɳʍɸɿʎɲʆɹʖʉʐʅɸɸʃɽɸʏɿ-
ʃɹʎʅɸɷɿɲʔʉʌɸʏɿʃɹʎɴɳʍɸɿʎʏʊʏɸͻ Ȱʆʏʉx " 3+
Ⱥɲɴɶɳɺʉʐʅɸʃʉɿʆʊʋɲʌɳɶʉʆʏɲʏɻʆɸʃɽɸʏɿʃɼʅɸʏɻʅɸɶɲʄʑʏɸʌɻɴɳʍɻȶʏʍɿɽɲɹʖʉʐʅɸ lim 0
xxa =
" 3+ɲʔʉʑ0 1< <a .
ͻ Ȱʆʏʉx " 3-ɽɲɴɶɳɺʉʐʅɸʃʉɿʆʊʋɲʌɳɶʉʆʏɲʏɻʆɸʃɽɸʏɿʃɼʅɸ ʏɻʅɿʃʌʊʏɸʌɻɴɳʍɻȶʏʍɿɽɲɹʖʉʐʅɸ lim 0
xxa =
" 3-ɲʔʉʑɽɲɸʀʆɲɿ 1>a .
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim ee2 3
5 2x x x
x x$++
" 3+.
ȶʖʉʐʅɸ
lim limee
e e
e e2 3
5 22 3
1 5 2
x x x
x x
x xx
xx
$++ =
+
+=
" "3 3+ + ````j jj j
lime
e2 3
1 5 2
31
x x
x
+
+=
" 3+ ``jj
.
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim ee2 3
4 2x x x
x x
1$
++
" 3- + .ȶʖʉʐʅɸ
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
.
ŝǀ ɇʏɲʊʌɿɲʅɸʄʉɶɲʌɿɽʅʃɹʎɸʔɲʌʅʊɺʉʐʅɸʏɻʆʀɷɿɲʅɸɽʉɷʉʄʉɶʀɲʅɸʏɿʎɸʃɽɸʏɿʃɹʎ
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim lnlnxx1x +" 3+.
ȶʖʉʐʅɸ
lim lnln lim
ln ln
ln limln
xx
x x
x
x1 1 1 1 1
11 01 1
x x x+ =+
=+
= + =" " "3 3 3+ + +` j .
ǀ Ʉʏɲʆɹʖʉʐʅɸʊʌɿʉʅɸʏʌɿɶʘʆʉʅɸʏʌɿʃɹʎɽɲʖʌɻʍɿʅʉʋʉɿʉʑʅɸʏɿʎɴɲ-ʍɿʃɹʎ ʏʌɿɶʘʆʉʅɸʏʌɿʃɹʎ ʍʖɹʍɸɿʎ x 1Ghn x 1Gvyo ʃɲɿ ɹʋɸɿʏɲɽɲɸʔɲʌʅʊɺʉʐʅɸʃʌɿʏɼʌɿʉʋɲʌɸʅɴʉʄɼʎ
Ʌʌʉʍʉʖɼ Ⱥɲʅʋʉʌʉʑʍɲʅɸɶɿɲʆɲʐʋʉʄʉɶʀʍʉʐʅɸʏɹʏʉɿɲʊʌɿɲʆɲʖʌɻ-ʍɿʅʉʋʉɿɼʍʉʐʅɸʏʉɽɸʙʌɻʅɲʏɻʎʅɻɷɸʆɿʃɼʎɸʋʀʔʌɲɶʅɹʆɻʎ
ʋʖ Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ lim xx
x
hn" 3+
.
14
ȶʖʉʐʅɸ xx
x x xx
x1 1 1+G G G
hn hn- .
lim x1 0
x- =
" 3+c m lim x
1 0x
=" 3+
Ȱʋʊʏʉʃʌɿʏɼʌɿʉʋɲʌɸʅɴʉʄɼʎʋʌʉʃʑʋʏɸɿʊʏɿ lim xx 0
x
hn =" 3+
.Ⱥɲʅʋʉʌʉʑʍɲʅɸʆɲʋʉʑʅɸɸʐʃʉʄʊʏɸʌɲʊʏɿʏʉ džhn ɸʀʆɲɿʔʌɲɶ-
ʅɹʆɻʍʐʆɳʌʏɻʍɻɈʉ lim x1 0
x=
" 3+ȵʋʉʅɹʆʘʎʏʉ lim x
x 0x
hn =" 3+
ʘʎʅɻ-ɷɸʆɿʃɼɸʋʀʔʌɲɶʅɹʆɻ
3.6 ɇʐʆɹʖɸɿɲɇʐʆɳʌʏɻʍɻʎɃʌɿʍʅʊʎ
ȶʍʏʘʅɿɲʍʐʆɳʌʏɻʍɻ f x] gʃɲɿx0 ɹʆɲʍɻʅɸʀʉʏʉʐʋɸɷʀʉʐʉʌɿʍʅʉʑʏɻʎȺɲʄɹʅɸʊʏɿɻf x] gɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉx0 ʊʏɲʆ lim f x f x
x x 00
="] ]g g.
Ƀʌɿʍʅʊʎɀɿɲʍʐʆɳʌʏɻʍɻ f x] gɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉʋɸɷʀʉʉʌɿʍʅʉʑʏɻʎʊʏɲʆ
ɸʀʆɲɿʍʐʆɸʖɼʎʍɸʃɳɽɸʍɻʅɸʀʉʏʉʐʋɸɷʀʉʐʉʌɿʍʅʉʑʏɻʎɇʐʆɸʖɸʀʎɸʃɸʀʋʉʐʉʌʀɺʉʆʏɲɿɸʀʆɲɿʉɿʋɲʌɲʃɳʏʘʍʐʆɲʌʏɼʍɸɿʎ
ͻ ȾɳɽɸʋʉʄʐʘʆʐʅɿʃɼʍʐʆɳʌʏɻʍɻP x] g.ͻ Ⱦɳɽɸʌɻʏɼʍʐʆɳʌʏɻʍɻ Q x
P x]] ggͻ Ƀɿʏʌɿɶʘʆʉʅɸʏʌɿʃɹʎʍʐʆɲʌʏɼʍɸɿʎ džhn džvyo džf ʃɲɿ džv .ͻ ȸɸʃɽɸʏɿʃɼ dža ʃɲɿɻʄʉɶɲʌɿɽʅɿʃɼlog xa ʅɸ0 1< !a .
ȺɸʙʌɻʅɲȰʆʉɿʍʐʆɲʌʏɼʍɸɿʎĨʃɲɿŐɸʀʆɲɿʍʐʆɸʖɸʀʎʍʏʉx0 ʏʊʏɸɸʀʆɲɿʍʐʆɸʖɸʀʎ
ʍʏʉx0 ʃɲɿʉɿʍʐʆɲʌʏɼʍɸɿʎ
f g+ c f$ f g$ gf Ĩ ʃɲɿ fv
ʅɸʏɻʆʋʌʉʒʋʊɽɸʍɻʊʏɿʉʌʀɺʉʆʏɲɿʍɸɷɿɳʍʏɻʅɲʋʉʐʋɸʌɿɹʖɸɿʏʉx0 .Ⱥɸʙʌɻʅɲ
ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ x0 ʃɲɿɻʍʐʆɳʌʏɻʍɻŐɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉf x0] gʏʊʏɸɻʍʑʆɽɸʍɼʏʉʐʎg f% ɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉx0 .ɀɸɽʉɷʉʄʉɶʀɸʎ
ŝ Ʉʏɲʆɽɹʄʉʐʅɸʆɲɷɸʀʇʉʐʅɸʊʏɿʅɿɲʃʄɲɷɿʃɼɸʀʆɲɿʍʐʆɸʖɼʎɽɲɸʄɹɶ-ʖʉʐʅɸʏɻʆʍʐʆɹʖɸɿɲʅɸʏʉʆʉʌɿʍʅʊʏɻʎʍʐʆɹʖɸɿɲʎʍʏʉʍɻʅɸʀʉɲʄʄɲ-ɶɼʎʏʑʋʉʐʏɻʎʍʐʆɳʌʏɻʍɻʎ
ŝŝ Ʉʏɲʆɽɹʄʉʐʅɸʆɲɲʋʉɷɸʀʇʉʐʅɸʊʏɿʅɿɲʍʐʆɳʌʏɻʍɻɸʀʆɲɿʍʐʆɸʖɼʎʍɸʊʄʉʏʉʋɸɷʀʉʉʌɿʍʅʉʑʏɻʎɼʍʚɹʆɲɷɿɳʍʏɻʅɲȴɲʌʃɸʀʆɲɲʋʉɷɸʀ-
15
ʇʉʐʅɸʊʏɿɸʀʆɲɿʍʐʆɸʖɼʎʍʚɹʆɲʏʐʖɲʀʉx0 ʏʉʐʋɸɷʀʉʐʉʌɿʍʅʉʑʏɻʎɼʏʉʐɷɿɲʍʏɼʅɲʏʉʎȴ
ŝŝŝȰʆʍɸʅɿɲɳʍʃɻʍɻʅɲʎɺɻʏɳʆʆɲɲʋʉɷɸʀʇʉʐʅɸʊʏɿɻʍʐʆɳʌʏɻʍɻɸʀ-ʆɲɿʍʐʆɸʖɼʎʍɸɹʆɲɷɿɳʍʏɻʅɲȴʃɲɿʅɲʎɷʀʆʉʐʆʅɿɲʍʐʆɲʌʏɻʍɿɲʃɼʍʖɹʍɻɶɿɲʏɻʍʐʆɳʌʏɻʍɻĨʃɲɿʊʏɿɸʀʆɲɿʍʐʆɸʖɼʎʍʚɹʆɲʍɻʅɸʀʉx1 ,ͻ ɽɲɶʌɳʔʉʐʅɸʏʉʆʉʌɿʍʅʊɶɿɲʏɻʆʍʐʆɹʖɸɿɲʍʚɹʆɲʏʐʖɲʀʉx0 .ͻ ɽɲʃɳʆʉʐʅɸʃɲʏɳʄʄɻʄɻɲʄʄɲɶɼʅɸʏɲɴʄɻʏɼʎʍʏʉʊʌɿʉ;ɲʆɳʄʉ-
ɶɲʅɸʏɻʍʐʆɲʌʏɻʍɿɲʃɼʍʖɹʍɻͿɹʏʍɿʙʍʏɸʆɲʋʌʉʃʑʗɸɿʏʉʊʌɿʉlim f xx x1"] gʏʉʉʋʉʀʉʅʋʉʌʉʑʅɸʆɲʏʉʐʋʉʄʉɶʀʍʉʐʅɸɸʋɸɿɷɼɶʆʘ-
ʌʀɺʉʐʅɸʊʏɿɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉx1 .
3.7 ȺɸʘʌɼʅɲʏɲɇʐʆɹʖɸɿɲʎȺɸʙʌɻʅɲ
ɀɿɲʍʐʆɳʌʏɻʍɻĨɽɲʄɹʅɸʊʏɿɸʀʆɲɿʍʐʆɸʖɼʎʍɸɹʆɲɲʆʉɿʖʏʊɷɿɳʍʏɻ-ʅɲ ,a b^ hʊʏɲʆɸʀʆɲɿʍʐʆɸʖɼʎʍɸʃɳɽɸʍɻʅɸʀʉʏʉʐ ,a b^ h.Ⱥɸʙʌɻʅɲ
ɀɿɲʍʐʆɳʌʏɻʍɻĨɽɲʄɹʅɸʊʏɿɸʀʆɲɿʍʐʆɸʖɼʎʍɸɹʆɲʃʄɸɿʍʏʊɷɿɳ-ʍʏɻʅɲ ,a b6 @ʊʏɲʆɸʀʆɲɿʍʐʆɸʖɼʎʍɸʃɳɽɸʍɻʅɸʀʉʏʉʐ ,a b^ hʃɲɿɸʋɿʋʄɹʉʆlim f x fx
a="a+] ]g gʃɲɿ lim f x f
xb=
"b-] ^g h.
ȺɸʙʌɻʅɲŽůnjĂŶŽȰʆ ʅɿɲ ʍʐʆɳʌʏɻʍɻ Ĩ ɸʀʆɲɿ ʍʐʆɸʖɼʎ ʍʏʉ ɷɿɳʍʏɻʅɲ ,a b6 @ ʃɲɿ
f f 0<$a b] ^g h ʏʊʏɸ ʐʋɳʌʖɸɿ ʏʉʐʄɳʖɿʍʏʉʆ ɹʆɲ ,x0 ! a b^ h ʏɹʏʉɿʉ ʙʍʏɸf x 00 =] g .Ʌɲʌɲʏɻʌɼʍɸɿʎ
ͻ Ɉʉ ɽɸʙʌɻʅɲŽůnjĂŶŽ ɸʇɲʍʔɲʄʀɺɸɿ ʏɻʆ ʑʋɲʌʇɻʅɿɲʎ ʏʉʐʄɳʖɿʍʏʉʆʄʑʍɻʎʏɻʎɸʇʀʍʘʍɻʎ f x 0=] g Ȱʐʏʊʍɻʅɲʀʆɸɿʊʏɿʅʋʉʌɸʀʆɲʐʋɳʌ-ʖʉʐʆʃɲɿʋɸʌɿʍʍʊʏɸʌɸʎ
ͻ ɈʉɽɸʙʌɻʅɲŽůnjĂŶŽɷɸʆɴʌʀʍʃɸɿʏɻʆʌʀɺɲʏɻʎɸʇʀʍʘʍɻʎ f x 0=] g ɲʋʄɳɸʇɲʍʔɲʄʀɺɸɿʏɻʆʑʋɲʌʇɻʅɿɲʎʏʉʐʄɳʖɿʍʏʉʆʌʀɺɲʎ
ͻ Ʉʏɲʆɷɸʆɿʃɲʆʉʋʉɿʉʑʆʏɲɿʃɲɿʉɿɷʑʉʋʌʉʒʋʉɽɹʍɸɿʎʏʉʐɽɸʘʌɼʅɲ-ʏʉʎŽůnjĂŶŽɷɸʆɸʀʆɲɿɲʋɲʌɲʀʏɻʏʉʆɲʐʋɳʌʖɸɿʌʀɺɲʏɻʎɸʇʀʍʘʍɻʎf x 0=] g .
ͻ ȰʆʅɿɲʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍɸɹʆɲɷɿɳʍʏɻʅɲȴʃɲɿɷɸʅɻ-ɷɸʆʀɺɸʏɲɿʍʚɲʐʏʊʏʊʏɸɲʐʏɼɼɸʀʆɲɿɽɸʏɿʃɼɶɿɲʃɳɽɸdž ! Dɼɸʀʆɲɿɲʌʆɻʏɿʃɼɶɿɲʃɳɽɸdž ! Dɷɻʄɲɷɼɷɿɲʏɻʌɸʀʍʏɲɽɸʌʊʋʌʊʍɻʅʉʍʏʉɷɿɳʍʏɻʅɲȴ
ͻ ɀɿɲʍʐʆɸʖɼʎ ʍʐʆɳʌʏɻʍɻ Ĩ ɷɿɲʏɻʌɸʀ ʍʏɲɽɸʌʊʋʌʊʍɻʅʉʍʏʉ ɷɿɳ-
16
ʍʏɻʅɲʋʉʐʉʌʀɺɸʏɲɿɲʋʊɷʑʉɷɿɲɷʉʖɿʃɹʎʌʀɺɸʎʏɻʎȵʋʉʅɹʆʘʎɶɿɲʆɲɴʌʉʑʅɸʏʉʋʌʊʍɻʅʉʏɻʎʍʐʆɳʌʏɻʍɻʎĨʍʚɹʆɲɷɿɳʍʏɻʅɲɽɲɴʌʀ-ʍʃʉʐʅɸʏʉʋʌʊʍɻʅʉʏɻʎʍʐʆɳʌʏɻʍɻʎʍɸɹʆɲʆɲʌɿɽʅʊʏʉʐɷɿɲʍʏɼ-ʅɲʏʉʎɲʐʏʉʑɈʉʋʌʊʍɻʅʉɲʐʏʊɸʀʆɲɿʃɲɿʏʉʋʌʊʍɻʅʉʏɻʎʍʐʆɳʌ-ʏɻʍɻʎĨʍʏʉɷɿɳʍʏɻʅɲɲʐʏʊ
Ʌʌʊʏɲʍɻȶʍʏʘ ʅɿɲ ʍʐʆɳʌʏɻʍɻ Ĩ ʍʐʆɸʖɼʎ ʍʏʉ ,a b^ h ʙʍʏɸ
lim limf x f x 0<x x
$" "a b+ -] ]g g Ɉʊʏɸ ʐʋɳʌʖɸɿ ʏʉʐʄɳʖɿʍʏʉʆ ɹʆɲ ,x0 ! a b^ h ʏɹʏʉɿʉ
ʙʍʏɸf x 00 =] g .ȸʋɲʌɲʋɳʆʘʋʌʊʏɲʍɻʅɲʎɷʀʆɸɿʏɻʆɷʐʆɲʏʊʏɻʏɲʆɲɸʔɲʌʅʊʍʉʐʅɸ
ʏʉɽɸʙʌɻʅɲʏʉʐŽůnjĂŶŽʍɸɲʆʉɿʖʏʊɷɿɳʍʏɻʅɲɼɻʅʀʃʄɸɿʍʏʉɷɿɳʍʏɻʅɲɅʌʊʏɲʍɻ
Ⱦɳɽɸʋʉʄʐʙʆʐʅʉʋɸʌɿʏʏʉʑɴɲɽʅʉʑɹʖɸɿʏʉʐʄɳʖɿʍʏʉʆʅɿɲʋʌɲɶʅɲ-ʏɿʃɼʌʀɺɲȺɸʙʌɻʅɲȵʆɷɿɳʅɸʍʘʆɈɿʅʙʆ
ȶʍʏʘʅɿɲʍʐʆɳʌʏɻʍɻĨ ɻʉʋʉʀɲɸʀʆɲɿʉʌɿʍʅɹʆɻʍɸɹʆɲʃʄɸɿʍʏʊɷɿ-ɳʍʏɻʅɲ ,a b6 @Ȱʆ
ͻ ɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,a b6 @ͻ Ĩ Ĩ!a b] ^g h
ʏʊʏɸɶɿɲʃɳɽɸɲʌɿɽʅʊʃʅɸʏɲʇʑʏʘʆ Ĩ a] gʃɲɿ Ĩ b^ hʐʋɳʌʖɸɿɹʆɲʎʏʉʐʄɳʖɿʍʏʉʆ ,x0 ! a b^ hʏɹʏʉɿʉʎʙʍʏɸf x0 l=] g .Ʌɲʌɲʏɼʌɻʍɻ
ȸɸɿʃʊʆɲĨ D] gɸʆʊʎɷɿɲʍʏɼʅɲʏʉʎȴʅɹʍʘʅɿɲʎʍʐʆɸʖʉʑʎʃɲɿʅɻʍʏɲ-ɽɸʌɼʎʍʐʆɳʌʏɻʍɻʎɸʀʆɲɿɷɿɳʍʏɻʅɲɇʏɻʋɸʌʀʋʏʘʍɻʋʉʐɻĨɸʀʆɲɿʍʏɲɽɸʌɼʏʉĨ D] gɸʀʆɲɿʅʉʆʉʍʑʆʉʄʉȺɸʙʌɻʅɲɀɹɶɿʍʏɻʎͲȵʄɳʖɿʍʏɻʎɈɿʅɼʎ
Ȱʆ ʅɿɲ ʍʐʆɳʌʏɻʍɻ Ĩ ɸʀʆɲɿ ʍʐʆɸʖɼʎ ʍʏʉ ʃʄɸɿʍʏʊ ɷɿɳʍʏɻʅɲ ,a b6 @ʏʊʏɸʐʋɳʌʖʉʐʆɷʑʉʏʉʐʄɳʖɿʍʏʉʆʍɻʅɸʀɲdžf ʃɲɿdžn ʏʉʐ ,a b6 @ʏɹʏʉɿɲʙʍʏɸʆɲɿʍʖʑɸɿf x f x f xG Gf n] ] ^g g hɶɿɲʃɳɽɸ ,x ! a b6 @ȴɻʄɲɷɼɻĨʋɲʀʌʆɸɿʍʏʉ,a b6 @ɸʄɳʖɿʍʏɻʏɿʅɼf xf] gʃɲɿʅɹɶɿʍʏɻʏɿʅɼf xn^ h.
Ʌɲʌɲʏɻʌɼʍɸɿʎͻ ȰʆɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,a b6 @ʏʊʏɸʏʉʍʑʆʉʄʉʏɿʅʙʆʏɻʎĨɸʀʆɲɿ
, ,f ma b L=^ h6 6@ @ʊʋʉʐŵɻɸʄɳʖɿʍʏɻʏɿʅɼʃɲɿɀɻʅɹɶɿʍʏɻʏɿʅɼʏɻʎ
ͻ ȰʆɻĨɸʀʆɲɿʍʐʆɸʖɼʎʃɲɿɶʆɻʍʀʘʎɲʑʇʉʐʍɲʍʏʉ ,a b6 @ʏʊʏɸʏʉʍʑʆʉ-ʄʉʏɿʅʙʆʏɻʎĨɸʀʆɲɿ , ,f f fa b a b=^ ] ^h g h6 6@ @.
17
ͻ ȰʆɻĨɸʀʆɲɿʍʐʆɸʖɼʎʃɲɿɶʆɻʍʀʘʎʔɽʀʆʉʐʍɲʍʏʉ ,a b6 @ʏʊʏɸʏʉʍʑ-ʆʉʄʉʏɿʅʙʆʏɻʎĨɸʀʆɲɿ , ,f f fa b b a=^ ^ ]h h g6 6@ @.
ͻ ȰʆɻĨɸʀʆɲɿʍʐʆɸʖɼʎʃɲɿɶʆɻʍʀʘʎɲʑʇʉʐʍɲʍʏʉ ,a b^ hʏʊʏɸʏʉʍʑʆʉ-ʄʉʏɿʅʙʆʏɻʎĨɸʀʆɲɿ , ,lim limf f x f x
x xa b =
" "a b+ -^^ ] ]`hh g gj.ͻ ȰʆɻĨɸʀʆɲɿʍʐʆɸʖɼʎʃɲɿɶʆɻʍʀʘʎʔɽʀʆʉʐʍɲʍʏʉ ,a b^ hʏʊʏɸʏʉʍʑ-
ʆʉʄʉʏɿʅʙʆʏɻʎĨɸʀʆɲɿ , ,lim limf f x f xx x
a b =" "b a- +^^ ] ]`hh g gj.
ͻ ɄʏɲʆɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,a b6 @ɹʖɸɿʅɹɶɿʍʏʉʃɲɿɸʄɳʖɿʍʏʉȶʍʏʘf xf] gʏʉɸʄɳʖɿʍʏʉʃɲɿf xn^ hʏʉʅɹɶɿʍʏʉʊʋʉʐ , ,x x ! a bf n 6 @Ȱʆʐʋʉ-ɽɹʍʉʐʅɸʊʏɿɸʀʆɲɿx x<f n ʏʊʏɸɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,x xf n6 @ɇʑʅ-ʔʘʆɲʅɸʏʉɽɸʙʌɻʅɲɸʆɷɿɳʅɸʍɻʎʏɿʅɼʎɶɿɲʃɳɽɸʃʅɸʏɲʇʑʏɻʎf xf] gʃɲɿf xn^ hʐʋɳʌʖɸɿɹʆɲʏʉʐʄɳʖɿʍʏʉʆ , ,x x! 3i a bf n^ ^h hʏɹʏʉɿʉʙʍʏɸf i l=] g ȵʋʉʅɹʆʘʎʊʏɲʆɹʖʉʐʅɸʍʐʆɳʌʏɻʍɻĨʍʐʆɸʖɼʍʏʉ ,a b6 @ɶɿɲʃɳɽɸʃʋʉʐɴʌʀʍʃɸʏɲɿʅɸʏɲʇʑʏɻʎʅɹɶɿʍʏɻʎʃɲɿʏɻʎɸʄɳʖɿʍʏɻʎʏɿʅɼʎʏɻʎĨʃɲɿʊʖɿʅɸʏɲʇʑʏʘʆĨ a] gʃɲɿĨ b^ hʐʋɳʌʖɸɿɹʆɲʏʉʐʄɳʖɿʍʏʉʆɽʙʍʏɸf i l=] g .
ɀɸɽʉɷʉʄʉɶʀɸʎŝ Ʉʏɲʆʍɸʅɿɲɳʍʃɻʍɻʅɲʎɷʀʆʉʐʆʅɿɲɸʇʀʍʘʍɻʏɻʎʉʋʉʀɲʎɲʋɲɿʏɸʀ-
ʏɲɿɻʑʋɲʌʇɻʅɿɲʎʏʉʐʄɳʖɿʍʏʉʆʌʀɺɲʎɽɲʅɸʏɲʔɹʌʉʐʅɸʏʉʐʎʊʌʉʐʎʍʏʉɹʆɲʅɹʄʉʎɽɲɽɹʏʉʐʅɸʍʐʆɳʌʏɻʍɻh x] gʃɲɿɽɲɸʔɲʌʅʊɺʉʐʅɸʏʉȺɸʙʌɻʅɲŽůnjĂŶŽ
ŝŝ Ȱʆɷɸʅʋʉʌʉʑʅɸʆɲʖʌɻʍɿʅʉʋʉɿɼʍʉʐʅɸɲʋɸʐɽɸʀɲʎʏɻʆʋʌʉɻɶʉʑ-ʅɸʆɻʅɸɽʉɷʉʄʉɶʀɲʋʖɶɿɲʏʀɻh x^ hɷɸʆʉʌʀɺɸʏɲɿɶɿɲʃɳʋʉɿɲʏɿʅɼʏʉʐdžʏʊʏɸɽɲʃɳʆʉʐʅɸʋʌʙʏɲɲʋɲʄʉɿʔɼʋɲʌʉʆʉʅɲʍʏʙʆɼɽɲɷɿ-ɲɿʌʉʑʅɸʅɸʃɳʋʉɿʉʆʋɲʌɳɶʉʆʏɲʃɲɿʅɸʏɳɽɲʅɸʏɲʔɹʌʉʐʅɸʏʉʐʎʊʌʉʐʎʍʏʉɹʆɲʅɹʄʉʎʃɲɿɽɲɽɹʏʉʐʅɸʍʐʆɳʌʏɻʍɻh x^ h.
ŝŝŝɀɸʌɿʃɹʎʔʉʌɹʎʅʋʉʌɸʀʆɲʖʌɸɿɲʍʏɸʀʆɲɸʋɿʄɹʇʉʐʅɸɹʆɲʅɿʃʌʊʏɸʌʉɷɿɳʍʏɻʅɲ ϭD ʐʋʉʍʑʆʉʄʉ ʏʉʐɲʌʖɿʃʉʑɷɿɲʍʏɼʅɲʏʉʎȴ ʋʉʐʅɲʎɴʉʄɸʑɸɿʃɲɿʆɲɸʔɲʌʅʊʍʉʐʅɸʏʉɽɸʙʌɻʅɲŽůnjĂŶŽʍɸɲʐʏʊȸʌʀɺɲʋʉʐɽɲɲʋʉɷɸʀʇʉʐʅɸʊʏɿ ʐʋɳʌʖɸɿ ɽɲɲʆɼʃɸɿ ʍʏʉ ϭD Ⱥɲɲʆɼʃɸɿʊʅʘʎʃɲɿʍʏʉȴɲʔʉʑʏʉ ϭ 3D D.
ŝǀ ȰʆʃɲʏɳʏɻʆɸʔɲʌʅʉɶɼʏʉʐɽɸʘʌɼʅɲʏʉʎŽůnjĂŶŽɶɿɲʏɻʍʐʆɸʖɼʍʐ-ʆɳʌʏɻʍɻĨʍʏʉ ,a b6 @ʋʌʉʃʑʗɸɿʊʏɿf f 0Ga b^ ^h h ʏʊʏɸɽɲʋɳʌʉʐʅɸɷʑʉʋɸʌɿʋʏʙʍɸɿʎͻ ɲʆf f 0<a b^ ^h h ʏʊʏɸʐʋɲʌʖɸɿɹʆɲʏʉʐʄɳʖɿʍʏʉʆ ,!p a b^ hʏɹʏʉɿʉ
ʙʍʏɸf 0p =^ h .
18
ͻ ɲʆ f f 0a b =^ ^h h ʏʊʏɸ f 0a =^ h ɼ f 0b =^ h ȴɻʄɲɷɼʏɲ ,a bɲʋʉ-ʏɸʄʉʑʆʌʀɺɸʎʏɻʎĨ;ɲʆʃɳʋʉɿʉɲʋʊʏɲɷʑʉɷɸʆʅʋʉʌɸʀʆɲɿʍʖʑɸɿʏʊʏɸɽɲɲʋʉʌʀʗʉʐʅɸʏɻʆɲʆʏʀʍʏʉɿʖɻʏɿʅɼͿȵʋʉʅɹʆʘʎp a= ɼp b= .
ȱʌɲʐʋɳʌʖɸɿɹʆɲʏʉʐʄɳʖɿʍʏʉʆ ,!p a b6 @ʏɹʏʉɿʉʙʍʏɸf 0p =^ h .ǀ ȳɿɲʆɲɲʋʉɷɸʀʇʉʐʅɸʊʏɿʅɿɲɸʇʀʍʘʍɻʏɻʎʅʉʌʔɼʎf x a=^ h ɹʖɸɿʅʀɲ
ʏʉʐʄɳʖɿʍʏʉʆʌʀɺɲɸʃʏʊʎɲʋʊʏʉɽɸʙʌɻʅɲŽůnjĂŶŽʅʋʉʌʉʑʅɸʆɲɴʌʉʑʅɸʏʉʍʑʆʉʄʉʏɿʅʙʆʏɻʎĨʃɲɿʆɲɲʋʉɷɸʀʇʉʐʅɸʊʏɿʋɸʌɿɹʖɸɿʏʉɲ
ǀŝȳɿɲʆɲɴʌʉʑʅɸʏʉʍʑʆʉʄʉʏɿʅʙʆʅɿɲʎʍʐʆɳʌʏɻʍɻʎ Ĩ ɽɲʃɳʆʉʐʅɸʅɸʄɹʏɻʅʉʆʉʏʉʆʀɲʎȳɿɲʃɳɽɸɷɿɳʍʏɻʅɲʍʏʉʉʋʉʀʉɻĨɸʀʆɲɿɶʆɻʍʀʘʎʅʉʆʊʏʉʆɻɽɲɴʌʀʍʃʉʐʅɸʏʉʍʑʆʉʄʉʏɿʅʙʆʏɻʎɶɿɲɲʐʏʊʏʉɷɿɳʍʏɻ-ʅɲɈʉʍʑʆʉʄʉʏɿʅʙʆʏɻʎĨɽɲɸʀʆɲɿɻɹʆʘʍɻʊʄʘʆʏʘʆʍʐʆʊʄʘʆʏɿʅʙʆʋʉʐɴʌʀʃɲʅɸʋʌʉɻɶʉʐʅɹʆʘʎ
3.8 ȵʌʘʏɼʍɸɿʎȺɸʘʌʀɲʎϭ ɁɲɷɿɲʏʐʋʙʍɸʏɸʏʉʃʌɿʏɼʌɿʉʋɲʌɸʅɴʉʄɼʎϮ Ɂɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ɶɿɲ ɹʆɲ ʋʉʄʐʙʆʐʅʉ P x^ h ɴɲɽʅʉʑ ʆ ɿʍʖʑɸɿ
limP x P xx x 0
0=
"^ ^h h.
ϯ Ɂɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ɶɿɲ ʏɻ ʌɻʏɼ ʍʐʆɳʌʏɻʍɻ f x Q xP x
=^ ^h hh ʊʋʉʐ,P x Q x^ ^h h ʋʉʄʐʙʆʐʅɲ ʏʉʐ dž ʃɲɿ x R0 ! ʅɸ Q x 00 !^ h ɿʍʖʑɸɿ
lim f x Q xP x
x x 0
0
0=
"^ ^h hh .
ϰ ɅʊʏɸʅɿɲʍʐʆɳʌʏɻʍɻĨʄɹɶɸʏɲɿʍʐʆɸʖɼʎʍʏʉʍɻʅɸʀʉx0 ʏʉʐʋɸɷʀʉʐʉʌɿʍʅʉʑʏɻʎ
ϱ ɅʊʏɸʅɿɲʍʐʆɳʌʏɻʍɻĨʄɹɶɸʏɲɿʍʐʆɸʖɼʎʍʚɹʆɲɷɿɳʍʏɻʅɲϲ ɁɲɲʋʉɷɸʀʇɸʏɸʊʏɿɲʆʅɿɲʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʉʌɿʍʅɹʆɻʃɲɿʍʐʆɸ-
ʖɼʎʍʏʉ ,a b6 @ʅɸ Ĩ Ĩ!a b] ^g hʏʊʏɸɶɿɲʃɳɽɸɲʌɿɽʅʊʃʅɸʏɲʇʑʏʘʆĨ a] g ʃɲɿ Ĩ b^ h ʐʋɳʌʖɸɿ ʏʉʐʄɳʖɿʍʏʉʆ ɹʆɲ ,x0 ! a b^ h ʏɹʏʉɿʉ ʙʍʏɸf x0 l=] g .
3.9 ȵʌʘʏɼʍɸɿʎʏʑʋʉʐɇʘʍʏʊͲȿɳɽʉʎϭ ȳɿɲʃɳɽɸʍʐʆɳʌʏɻʍɻĨʋʉʐʉʌʀɺɸʏɲɿʍʏʉR ɸʀʆɲɿ lim f 0
x 0=
"^ h.
Ϯ ȰʆɶɿɲʏɿʎʍʐʆɲʌʏɼʍɸɿʎĨ ŐʃɲɿŚɿʍʖʑʉʐʆͻ h x f x g xG G] ] ]g g gʃʉʆʏɳʍʏʉx0 ,ͻ limh x l
x x 10
="] g ʃɲɿ limg x l
x x 20
="] g ʊʋʉʐ ,l l R1 2 ! ,
ʏʊʏɸ liml f x lx x1 2
0G G
"] g .
19
ϯ Ȱʆf x 0>] g ɶɿɲʃɳɽɸdž R! ʃɲɿʐʋɳʌʖɸɿʏʉ lim f xx 0" +] gʏʊʏɸʃɲʏʚɲʆɳ-
ɶʃɻɿʍʖʑɸɿ lim f x 0>x 0" +] g .
ϰ Ȱʆɿʍʖʑɸɿlim f x 0x 0
=" +] g ʏʊʏɸɷɸʆɿʍʖʑɸɿʃɲʏʚɲʆɳɶʃɻʊʏɿlim f x
1x 0
3=+" + ] g .
ϱ Ȱʆ lim f x 0x x0
=" +] g ʏʊʏɸ lim f x
1x x0
3=+" + ] g .
ϲ Ȱʆɿʍʖʑɸɿ lim f xx 0
2 3=+"^ h ʏʊʏɸ lim f x
x 03=+
"^ h ɼ lim f x
x 03=-
"^ h .
ϳ Ȱʆʐʋɳʌʖɸɿʏʉ lim f x g xx 0
$"^ ^^ h hhʏʊʏɸɸʀʆɲɿʀʍʉʅɸf g0 0$^ ^h h.
ϴ Ȱʆ f x 0<^ h ɶɿɲʃɳɽɸ dž R! ʃɲɿɻʍʐʆɳʌʏɻʍɻĨɹʖɸɿʊʌɿʉʍʏʉʍɻ-ʅɸʀʉϬɽɲɸʀʆɲɿ lim f x 0<
x 0"^ h .
ϵ Ȱʆ lim f x 0x 2
="^ h ʃɲɿ limg x
x 23=+
"^ h ʏʊʏɸ lim g x
f x0
x 2=
" ^ hh .
ϭϬ ȸɸɿʃʊʆɲĨ D] gɸʆʊʎɷɿɲʍʏɼʅɲʏʉʎȴʅɹʍʘʅɿɲʎʍʐʆɸʖʉʑʎʍʐʆɳʌʏɻ-ʍɻʎĨɸʀʆɲɿɷɿɳʍʏɻʅɲ
ϭϭ ȾɳɽɸʋʉʄʐʘʆʐʅɿʃɼʍʐʆɳʌʏɻʍɻɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉR .ϭϮ ȰʆʅɿɲʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿɲʍʐʆɸʖɼʎʍɸʃɳʋʉɿʉʍɻʅɸʀʉʏʉʐʋɸɷʀʉʐ
ʉʌɿʍʅʉʑʏɻʎʏʊʏɸʍʚɲʐʏʊʏʉʍɻʅɸʀʉɷɸʆɽɲʐʋɳʌʖɸɿʏʉʊʌɿʊʏɻʎϭϯ Ⱦɳɽɸʍʐʆɳʌʏɻʍɻɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉʋɸɷʀʉʉʌɿʍʅʉʑʏɻʎϭϰ ȰʆɶɿɲʏɿʎʍʐʆɲʌʏɼʍɸɿʎĨʃɲɿŐʋʉʐʉʌʀɺʉʆʏɲɿʍʏʉR ɻŐɸʀʆɲɿʍʐ-
ʆɸʖɼʎʍʏʉR ʃɲɿɿʍʖʑɸɿf x g x2 2=] ]g gɶɿɲʃɳɽɸdž R! ʏʊʏɸɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉR .
ϭϱ ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʚɹʆɲɷɿɳʍʏɻʅɲȴʏʊʏɸʉʌʀɺɸʏɲɿʃɲɿɻʍʑʆɽɸʍɻĨ Ĩ% ʍʏʉȴʃɲɿɸʀʆɲɿʍʐʆɸʖɼʎʍɸɲʐʏʊ
ϭϲ ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉR ʏʊʏɸɻɸʇʀʍʘʍɻ f x 0=^ h ɹʖɸɿʅɿɲʏʉʐʄɳʖɿʍʏʉʆʌʀɺɲ
ϭϳ ȸɸɿʃʊʆɲĨ D] gɸʆʊʎɲʆʉɿʃʏʉʑɷɿɲʍʏɼʅɲʏʉʎȴʅɹʍʘʅɿɲʎʍʐʆɸʖʉʑʎʃɲɿʅɻʍʏɲɽɸʌɼʎʍʐʆɳʌʏɻʍɻʎĨɸʀʆɲɿɲʆʉɿʃʏʊɼʃʄɸɿʍʏʊɷɿɳʍʏɻʅɲ
ϭϴ ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉR ʙʍʏɸf f f 0<a b c] ^ ^g h h ʅɸ, , R!a b c ,ʏʊʏɸʐʋɳʌʖɸɿx R0 ! ʙʍʏɸf x 00 =] g .
ϭϵ ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,a b6 @ʙʍʏɸ f xG Ga b] g ɶɿɲ
ʃɳɽɸ ,x ! a b6 @ʏʊʏɸʐʋɳʌʖɸɿ ,x0 ! a b6 @ʙʍʏɸf x 20a b= +] g .
ϮϬ ȶʍʏʘ ɻ ʍʐʆɸʖɼʎ ʍʐʆɳʌʏɻʍɻ :f R R" ɶɿɲ ʏɻʆ ʉʋʉʀɲ ɿʍʖʑɸɿ
x f x x x3 H hn-] g ɶɿɲʃɳɽɸdž R! Ɉʉf 0] gɸʀʆɲɿ 61 .Ϯϭ ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,0 16 @ʃɲɿɻɸʇʀʍʘʍɻ f x 0=^ h
ɹʖɸɿʅʀɲʌʀɺɲx0 ʍʏʉɷɿɳʍʏɻʅɲ ,0 1^ hʏʊʏɸɽɲɿʍʖʑɸɿf f0 1 0<^ ^h h .ϮϮ ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,a b6 @ʏʊʏɸɻĨɹʖɸɿɹʆɲʅʉʆɲ-
ɷɿʃʊʅɹɶɿʍʏʉʃɲɿɹʆɲʅʉʆɲɷɿʃʊɸʄɳʖɿʍʏʉʍʏʉ ,a b6 @.
20
Ϯϯ Ȱʆ Ĩ ʍʐʆɸʖɼʎ ʃɲɿ ɶʆɻʍʀʘʎ ʅʉʆʊʏʉʆɻ ʍʐʆɳʌʏɻʍɻ ʍʏʉ ,a b6 @ ʅɸf x 00 =^ h ,x0 ! a b^ hʏʊʏɸf f 0<a b^ ^h h .
Ϯϰ Ȱʆʅɿɲʍʐʆɳʌʏɻʍɻ : ,f R"a b6 @ ɸʀʆɲɿʍʐʆɸʖɼʎʏʊʏɸʏʉʍʑʆʉʄʉʏɿ-ʅʙʆʏɻʎɸʀʆɲɿʃʄɸɿʍʏʊɷɿɳʍʏɻʅɲ
Ϯϱ Ȱʆ ɻ ʍʐʆɳʌʏɻʍɻ Ĩ ɸʀʆɲɿ ʍʐʆɸʖɼʎ ʍʏʉ ɷɿɳʍʏɻʅɲ ,a b6 @ ʅɸĨ Ĩ!a b^ ^h h,ʏʊʏɸɻĨʋɲʀʌʆɸɿʏɿʅɹʎʅʊʆʉʅɸʏɲʇʑʏʘʆĨ a^ hʃɲɿĨ b^ h.
3.10 Ȱʍʃɼʍɸɿʎɶɿɲʄʑʍɻϭ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ limx
x e2 51π
x
x
2 5
2004 3hn
+- +
"-
+^^ h h ɴ lim
x x xx x e6 9 16 8
x
x
3 2
2 3
- + + +- + +
"
-
.
ɶ lim x x x2 1x 2
3 2- + -"-^ h ɷ lim x x x2 3 1 13 4
x 2
25 23- + + -"^ h7 A.
ɸ lim x xx x
5 67 11
x 1 2
2
- ++ -
" ʍʏ lim x
x x1
3 30x 2 2
5
-+ -
".
Ϯ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx2
4x 2
2
+-
"- ɴ lim
x
x2 14 1
x 21
2
-
-"
.
ɶ lim x xx
22 8
x 0 2
3
-+ -
"
^ h ɷ lim x x
x x x3 10
4 5 2x 2 2
3 2
+ -- + -
".
ɸ lim x xx
x3 21
44
x 2 2 2- +- - -"
a k ʍʏ lim x x xx
11
x 1 3 2
3
+ + ++
"-.
ϯ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx x
13 4
x 1 2
2
-+ -
" ɴ lim x x x
x x3 6 22 1
x 1 3 2
4 2
+ - +- +
".
ɶ lim x xx x
4 33 2
x 1 4
3
- +- +
" ɷ lim x x
x5 416
x 4 2
2
- +-
".
ɸ lim xx x
12 3 1
x 1
2
-- +
" ʍʏ lim x x
x x33
x 0 2
2
+-
".
ϰ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ limxx
24
x 4 --
" ɴ lim
x xx x
3 3 13 4
x 1 2
2
- + -+ -
" ɶ lim
xx
5 33 2 2
x 2 2 + -- -
".
ɷ lim xx1 1
x 0
+ -"
ɸ lim xx x1 1
x 0
- - +"
ʍʏ lim xx49
2 3x 7 2 -
- -"
.
ϱ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ limx
x1 2
1x 1 3 3
2
- --
"- ɴ lim
xx
28
x 8 3 --
".
ɶ limx
x x1
3 3 1x 1 3 -
+ - +"
ɷ lim xx x
113 8
x 1 -+ - + +
".
21
ɸ lim x xx3 2
4x 2 3 - +
-"
ʍʏ limx
x x11 1
x 1 2 -+ - -
".
ϲ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim x xx x x
2 1 12 3 1 2
x 0 2
2
- + - -- - - + -
" ɴ lim x
x x x4
8 16 4x 4
2
-- + - -
".
ɶ lim xx x x
12 3 1
x 1 2
2
-- + - +
" ɷ lim x x
x x3 2
2 3 12x 2 2
2
- ++ + -
".
ɸ lim x xx x
7 62 2
x 1 2 - +- + -
" ʍʏ lim x
x x x9
1 14x 3 2
2
-- + + -
".
ϳ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx x x
11 3
x 1 2
2
-- + +
"- ɴ lim x
x x1
1 1x 1
2
-- + -
".
ɶ lim xx x x
24 5 6
x 2
2 2
-- + - +
" ɷ lim x
x x1
2 1x 1
2
-- +
".
ɸ lim xx x x
11 3 2
x 1 2
2
-- + - +
" ʍʏ lim x x
x x5 6
2 4x 2 2
2
- +- + -
".
ϴ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim x xx x
9 82 2 4 7
x 1 2 + ++ + - -
"- ɴ lim x x
x x x2 7 4
6 4 1 4 2x 4 2 + -
+ + + + +"-
.
ɶ lim x xx x
22 1 1
x 2 2 ++ +- +
"- ɷ lim x x
x x3
3 3x 3 2 -
- + -"
.
ɸ lim xx xx x3
x 0 2
3
++ -
" ʍʏ lim x
x x x1
10 9 1x 1
2
-- + - -
".
ϵ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx x x2
x 0
2 + -"
ɴ lim x xx x5 4
4 2 8x 4 2 - +
- + -"
.
ɶ lim xx x x
93 3
x 3 2
2
-- + -
" ɷ lim x x
x x1 2 3x 1 2
3
+ --
".
ɸ lim xx x x x
22 5 6
x 2
2 2
-- - - +
" ʍʏ lim x x
x x x7 6
2 1 2 2x 2 2
3
- +- + + -
".
ϭϬ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx
23
x 0
hn"
ɴ lim xx
42
x 0 ff
" ɶ lim x
x2x 2r
vyo-" r
.
ɷ lim x xx x
x 0 hnhn
+-
" ɸ lim x
x x1 3 1 2x 0
hn hn+ - -"
.
ʍʏ limxx
3 9x 0
hn- +"
.
ϭϭ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx
x 0
f"
ɴ lim xx x
x 0
2
hn-
" ɶ lim x
x x3x 0
2
f+
".
22
ɷ lim x xx x
42
x 0 2f hn
+-
" ɸ lim x
xx 0
hn"
ʍʏ lim xx
x 0
hn"
.
ϭϮ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx1 1
x 0
hn- +"
ɴ limx
x1 2x 0 vyo-"
.
ɶ lim x xx x x
x 0 2
4 2 hn+
+ -"
ɷ lim x xx x
5 73 4
x 0 hnhn
+-
".
ɸ limx x
x3 3
1 2x 0
vyo+ -
-" ^ h ʍʏ lim x
x1x 0 2
vyo-"
.
ϭϯ Ɂɲɴʌɸɽʉʑʆɲʆʐʋɳʌʖʉʐʆʏɲʊʌɿɲʏʘʆ
ɲ,,
f xx x x
x x3 5 4
2 4π>2
Gvyo=
- --
^ ^h h) ʍʏʉx 40 = .
ɴ,
,f x x x
x x x
xx x x x
3 26 8 2
24 4 2 2
>
<
2
2
2
hn
= - +- +
-- + + -
^^
hhZ
[
\
]]
]]ʍʏʉx 20 = .
ɶ,
,f x x
x x x
x x31 3
5 3
<2
4
2 H= +
- +
-^ h * ʍʏʉx 30 = ʃɲɿʍʏʉx 30 =- .
ϭϰ Ɂɲɴʌɸɽɸʀʏʉ R!a ʙʍʏɸɶɿɲʏɻʍʐʆɳʌʏɻʍɻ,,
f xx xx x2 1 0
2 3 0>Ga
a=
+ -+ -
^ h ( ʆɲʐʋɳʌʖɸɿʏʉʊʌɿʉ lim f x
x 0"^ h.
ϭϱ Ɂɲ ɴʌɸɽʉʑʆ ʏɲ , R!a b ʙʍʏɸ ɶɿɲ ʏɻ ʍʐʆɳʌʏɻʍɻ,
,,
f xx x
xx x
00 0
1 0
<
>
a b
b a
=+
=+ -
^ hZ
[
\
]]
]ʆɲɿʍʖʑɸɿ lim f x 1
x 0=
"^ h .
ϭϲ Ɂɲ ɴʌɸɽʉʑʆ ʏɲ , R!a b ʙʍʏɸ ɶɿɲ ʏɻ ʍʐʆɳʌʏɻʍɻ,,
f xx x x
x x x1 213 2>
2
2
Ga
b=
+ +- +
^ h ) ʆɲʐʋɳʌʖɸɿʏʉʊʌɿʉ lim f xx 2"^ hʃɲɿɻɶʌɲʔɿ-
ʃɼʋɲʌɳʍʏɲʍɻʏɻʎĨʆɲɷɿɹʌʖɸʏɲɿɲʋʊʏʉ ,M 1 3-^ h.ϭϳ Ɂɲ ɴʌɸɽʉʑʆ ʏɲ , R!a b ʙʍʏɸ ɶɿɲ ʏɻ ʍʐʆɳʌʏɻʍɻ
, ,, ,
, ,f x
x xx x xx x
1 12 1 43 2 4 5
3
2
!
!
!
a
a b
b
=+ -+ ++
^ ^^
h6 @@@
Z
[
\
]]
] ʆɲʐʋɳʌʖʉʐʆʏɲʊʌɿɲ lim f x
x 1"^ h ʃɲɿ
lim f xx 4"^ h.
ϭϴ Ɂɲɴʌɸɽʉʑʆʉɿʏɿʅɹʎʏʘʆ , R!a b ʙʍʏɸʆɲɸʀʆɲɿlim xx x
4 2x 2 2
2 a b-
- + ="
.
23
ϭϵ Ɂɲɴʌɸɽɸʀʏʉʊʌɿʉ lim f xx x0"^ hʊʏɲʆ
ɲ lim f x x4 3 1x 2
+ - ="^^ h h ɴ lim x
f x3
2 1 4x 3 -
- ="
^ h.
ɶ limx x
f x1 1
2x 3 + - +
="
^ h ɷ lim x
f x4
3 5 5x 4 -
- ="
^ h.
ϮϬ Ȱʆ x f x x3$ Hhn hn^ h ɶɿɲʃɳɽɸdž R! ʃɲɿ lim f x Rx 0
!l="^ h ʆɲɴʌɸ-
ɽɸʀʉl.Ϯϭ Ɂɲɴʌɸɽʉʑʆʏɲʊʌɿɲʏɻʎf x^ hʍʏʉx0 ʊʏɲʆ
ɲ xx f x x x
x24
212 242
2G G--
- --^ h x 20 = .
ɴ xx f x x x
x11
4 31
2G G--
- +-^ h x 10 = .
ɶ x xx x x f x x x
x2 34 3
3 10 32 9
2
3 2
2
2
G G+ -+ +
+ +-^ ^h h
x 30 =- .
ɷ xx f x x
x1 2 1G Ghnvyo vyo- -^ h x 00 = .
ɸ x xf x xG Ghn f^ h x 00 = .ϮϮ Ȱʆɿʍʖʑɸɿ x f x x4 42G G +^ h ɶɿɲʃɳɽɸdž R! ʆɲɴʌɸɽʉʑʆʏɲʊʌɿɲ
ɲ lim xf x
28
x 2 --
"
^ h ɴ lim x
f x x22
x 2
2
--
"
^ h.
Ϯϯ Ȱʆɶɿɲʏɻʍʐʆɳʌʏɻʍɻf x^ hɿʍʖʑɸɿ x f x f x x4 5 2 2 32 2G G- - -^ ^h h ɶɿɲʃɳɽɸdž R! ʆɲɴʌɸɽɸʀʏʉ lim f x
x 1"^ h.
Ϯϰ Ȱʆ lim g xf x
Rx x0
!a=" ^ hh ʃɲɿ f x g x f x g x3 43 2 3+ =^ ^ ^ ^h h h h ɶɿɲ ʃɳɽɸ
dž R! ʆɲɴʌɸʀʏɸʏʉɲ.Ϯϱ Ȱʆ lim f x 0
x 2=
"^ h ʆɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ limf x
f x1 1x 2
hn
+ -" ^^^hhh
ɴ lim f xf x f x
2 22 1 1
x 2
2
- -- + -
" ^^ hh h.
Ϯϲ Ȱʆ lim f x x x2 4 2 5x 3
2+ - - ="^^ h h ʆɲʐʋʉʄʉɶɿʍʏɸʀʏʉ lim f x
x 3"^ h.
Ϯϳ Ȱʆɿʍʖʑɸɿx f x f xa+ = -^ ^h hɶɿɲʃɳɽɸdž R! ʅɸlim f x x2 1x
a+ - ="a^^ h h
ʆɲɴʌɸɽɸʀʏʉ lim f xx 0"^ h.
Ϯϴ Ȱʆ lim xf x 1
x 0=
"
^ hʆɲɴʌɸʀʏɸʏʉ lim
xf x11
x 1 --
"
^ h.
Ϯϵ Ȱʆlim lnf x x x1 3 3x 1
2- + - + ="^^ h h ʆɲʐʋʉʄʉɶɿʍʏɸʀʏʉlim f x
f x1x 0 2
hn r-" ^ ^^ h hh .
ϯϬ Ȱʆ lim xf x x 1 2
x 0
hn+ -=
"
^ hʆɲʐʋʉʄʉɶɿʍʏɸʀʏʉ lim
f x xf x 1
x 0 vyo--
" ^^hh
.
24
ϯϭ Ȱʆ f x f x x x x1 2 4 22- + = - +^ ^^h h h ɶɿɲ ʃɳɽɸ dž R! ʃɲɿ
lim xf x f
11
x 1l-
- ="
^ ^h hʏʊʏɸʆɲɴʌɸɽɸʀʉ 0>l .
ϯϮ Ȱʆlim xf x
4 31
x 2 2 - ="
^ hʃɲɿlim g x x 8 3
2x 2
3 - =-"^ ^^ h hh ʆɲɴʌɸɽɸʀʏʉlimh x
x 2"^ h
ʊʋʉʐh x f x g x=^ ^ ^h h h.ϯϯ Ȱʆlim x f x1 1 4
x 0+ - =
"^ ^_ h hi lim g x x 1 2
x 0vyo - =
"^ ^^ h hh ʆɲɴʌɸʀʏɸʏʉ
lim g xf x
x 0" ^ hh .ϯϰ Ȱʆ lim f x g x3 19
x 3+ =
"^ ^^ h hh ʃɲɿ lim f x g x2 3
x 3- =-
"^ ^^ h hh ʆɲɴʌɸʀʏɸ ʏɲ
lim f xx 3"^ hʃɲɿ limg x
x 3"^ h.
ϯϱ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx x
48 16
x 4 4
2
-- +
" ^ h ɴ limx xx8 163
x 4 2 - +-
"
ɶ lim x xx2 1
2 3x 1 2 - +
+"
ɷ limx x x
x3 3 1
1x 1 3 2
2
+ + +-
"-.
ɸ limx
x xx 0 54
$hn vyo"
ʍʏ limxx
x 0 43
hn"
.
ϯϲ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim x x xx
3 3 15 6
x 1 3 2- + --
" ɴ lim x x x
x4
22x 2 2 2- - -"
a k.ɶ lim x
xx x1 2
2x 1 2 2- - + -"a k ɷ lim x
xxx
12 1
1x 1 -- - -"
c m.ɸ lim x x
1 1x 0
-"c m ʍʏ lim x x
x x3 5x 0
2 + -"
.
ϯϳ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ limx x x
x1
1x 1 3 2- - +
-"
ɴ limx xx
25
x 4 - --
".
ɶ limx x x x2
32x 4 - +
--"
ɷ lim x x21
21
x 2 - - -"c m.
ɸ limxx
13 5
x 1 --
" ʍʏ lim x
x1x 0 2
vyo-"
.
ϯϴ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx
12
x 2hn -
-" r
ɴ lim xx 1
x 0 hn-
".
ɶ lim xx
13
x vyo +-
"r ɷ lim x
x1
1x 0 vyo -
-"
.
25
ɸ lim xx
2 11
x 3vyo -
-" r
ʍʏ lim xx
2 13
x 67 hn +
-" r
.
ϯϵ ȳɿɲʏɿʎɷɿɳʔʉʌɸʎʏɿʅɹʎʏʉʐ R!a ʆɲʐʋʉʄʉɶɿʍʏʉʑʆɲʆʐʋɳʌʖʉʐʆʏɲʊʌɿɲ
ɲ lim x x x xx x4 6 4 1
2x 1 4 3 2
2 2a a- + - +
- +"
ɴ lim xx x
22
x 2
2 a a-
- +"
.
ɶ lim x x xx x3 3 1
5x 1 3 2
2 a- + -+ -
"
ϰϬ ȳɿɲʏɿʎɷɿɳʔʉʌɸʎʏɿʅɹʎʏʘʆ , R!a b ʆɲʐʋʉʄʉɶɿʍʏʉʑʆɲʆʐʋɳʌ-ʖʉʐʆʏɲʊʌɿɲ
ɲ lim x xx x2
x 0
2
hna b+ +
" ɴ lim x x
x x4 4
2 4x 2 2
2 b a- ++ +
".
ɶ lim xx x 1
x 0 2a b+ - +
"
ϰϭ Ɂɲɴʌɸɽɸʀ
ɲ ʏʉ lim f xx 1"^ hʊʏɲʆ lim x
x f x1
1x 1 3 3-
- =+"
^ ^h h.
ɴ ʏʉ lim f xx 0"^ hʊʏɲʆ lim f x
xx
23x 0
3hnf
=+" ^ h .
ɶ ʏʉlim f xx 4"^ hɲʆɶɿɲʃɳɽɸx 4! ɿʍʖʑɸɿf x x x x
x x12 48 64
5 43 2
2
H - + -- +^ h .
ɷ ʏʉ lim f xx 1"^ hɲʆɶɿɲʃɳɽɸx 1! ɿʍʖʑɸɿf x x x x
x x3 3 15 4
3 2
2
G - + -- +^ h .
ɸ ʏʉ lim f xx 3"^ hɲʆɶɿɲʃɳɽɸx 3! ɿʍʖʑɸɿf x x
x3
1 2H -
+ -^ h .
ϰϮ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲɲ lim x x x3 2 5
x3 2- + -
" 3+^ h ɴ lim x x x3 5 2 5
x3 4- + -
" 3+^ h.
ɶ lim x x x3 2 7x
4 3- + -" 3-^ h ɷ lim x x2 4 5
x3 2- +
" 3-^ h.
ɸ lim x x x2 5 6x
3 2+ - -" 3-
ʍʏ lim x x x2 3 7x
5 2- + +" 3-^ h
ϰϯ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim x xx x2 1
5 2x 3
4 2
+ -- +
" 3- ɴ lim x
x x1
4 4 6x 2
3 2
-- +
" 3+.
ɶ lim x xx x
6 3 112 5 6
x 6
6 2
+ -+ -
" 3- ɷ lim x x
x x x2 1
4 2x 5 3
5 3
- +- + -
" 3-.
ɸ lim x xx x
3 19 16 7
x 5 2
3
- -+ -
" 3- ʍʏ lim x
x x2 7
4 5 2x 100
99
++ +
" 3-.
ϰϰ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim x xx x x x x
4 5 82 5 1 7
x 2
3 2 3
- ++ - + + -
" 3- ɴ lim x
xx
x x11
2 3x
2 2- - +- -
" 3+c m.
26
ɶ lim x x2 1x
23 + -" 3+
ɷ lim x x7 5 3x
2 + -" 3-
.
ɸ lim x x x x2 4 2 1x
2 5 25+ - + - + +" 3-^ h.
ʍʏ lim x x x1x
2 - + -" 3-^ h.
ϰϱ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim x x x2x
2 + +" 3-^ h ɴ lim x
x x2 5
4 4x
2
-+ +
" 3-.
ɶ lim x x x8 1x
33 + - -" 3+^ h ɷ lim x x x5 3
x4 + - +
" 3-^ h.
ɸ lim x x x4 1 2x
2 + - -" 3+^ h ʍʏ lim x x x1
x2 + + -
" 3+^ h.
ϰϲ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲɲ lim x x x1
x2 + + +
" 3-^ h ɴ lim x x x9 2 3
x2 + + +
" 3-^ h.
ɶ lim x x x1x
2 2+ +" 3-^ h ɷ lim x x x x1 4
x2 2+ + - -
" 3-^ h.
ɸ lim x x x x4 1 1 3x
2 2+ + + + -" 3+^ h.
ʍʏ lim x x x x9 2 1 1 2x
2 33+ + - + -" 3+^ h.
ϰϳ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ limx x xx x x
1 11 1
x 2 2
4 4
+ - + ++ + - +
" 3- ɴ lim x x 1
x3 3- -
" 3-^ h.
ɶ limxx
xx11x
2 +- +
-" 3+
ɷ lim x x x xx
+ - -" 3+_ i.
ɸ lim x x x xx
+ + -" 3+_ i.
ʍʏ lim x x x x x4 1 3x
2 2+ + + + -" 3+^ h.
ϰϴ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim ee e3 2
1x x
x x2
++ -
" 3+
+
ɴ lim 1 2 66 3 6 3
x x
x x
2
1
$$
-- +
" 3+ +
+
.
ɶ lim 2 52 1
x x
x 1
+-
" 3+
+
ɷ lim 2 2 12 2 3
x x x
x x
3
1 2
+ -+ -
" 3- +
- +
.
ɸ lim e ee e
12
x x x
x x
1 1
2
+ -+ -
" 3+ - +
+
ʍʏ limee
31
x x
x
01
1
++
" +.
ϰϵ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim ee
33
x x x
x x
1
1
+-
" 3+ +
+
ɴ lim ee
22
x x x
x x
2
1 2
++
" 3+ +
+ +
.
ɶ lim lnlnxx1x +" 3+ ɷ lim ln
lnxx1x +" 3-.
ɸ lim ee
3 22 3
x x x x
x x x
1 1 2
2 1
+ -- +
" 3+ + - -
+ -
ʍʏ lim ee
3 22 3
x x x x
x x x
1 1 2
2 1
+ -- +
" 3- + - -
+ -
.
27
ϱϬ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲɲ lim x x 5
xx6 + -
" 3-^ h ɴ lim x2 3
xx x 6- -
" 3+-^ h.
ɶ lim x xe
12
x
x
2+ ++
" 3- ɷ limln x x1
x+ -
" 3+^ h.
ɸ limln x xex
x 0 2hn+"
a k ʍʏ lim1 21
x x01
+" +.
ϱϭ Ɂɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʋɲʌɲʃɳʏʘʊʌɿɲ
ɲ lim xx
x
hn" 3+
ɴ lim xx
x
vyo" 3-
.
ɶ limx x1
xhn
" 3+ ɷ limx x
1x 0hn
".
ɸ lim x xx x x3 2
2 3x
2
vyohn
+-
" 3- ʍʏ lim x
xx1
x 0
2
hn hn".
ϱϮ ȳɿɲʏɿʎɷɿɳʔʉʌɸʎʏɿʅɹʎʏʘʆ R!a ʆɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʊʌɿɲ
ɲ lim x xx x
21
3 72 3
x 2
3
aa--
+ -- +
" 3+ ^ hh ɴ lim x x xx x x45 2
8x
4 2
3 2aa-
- + -+ - +" 3- ^ h .
ɶ lim x x x3 2 1x
2 a+ + - +" 3-^ h ɷ lim x x x3
x2 a+ + +
" 3-^ h.
ϱϯ ȳɿɲʏɿʎɷɿɳʔʉʌɸʎʏɿʅɹʎʏʘʆ 0>a ʆɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʊʌɿɲ
ɲ lim 3 13 1
x x x
x x
1
1
aa
+ -+ +
" 3+ -
+
ɴ lim 22 4
x x x
x x
1
1
aa+
-" 3+ -
+
.
ϱϰ ȳɿɲʏɿʎɷɿɳʔʉʌɸʎʏɿʅɹʎʏʘʆ , R!a b ʆɲʐʋʉʄʉɶɿʍʏʉʑʆʏɲʊʌɿɲ
ɲ lim x xx x
11
x 3
2
a ba a b
- +- - +
" 3+
^ h ɴ lim x x
x x3
1 2 1x 2
2
ba
+ -- + -
" 3+
^ h.
ϱϱ ȴʀʆɸʏɲɿɻʍʐʆɳʌʏɻʍɻ : ,f 1 R"3+^ h ɶɿɲʏɻʆʉʋʉʀɲɿʍʖʑɸɿ
xx x f x x
x2
9 3 1 3 42
G G+- + +^ h ɶɿɲʃɳɽɸ ,x 1 3! +^ h.
Ɂɲɴʌɸɽɸʀʏʉ lim f xx" 3+^ h.
ϱϲ ȶʍʏʘʍʐʆɳʌʏɻʍɻĨʉʌɿʍʅɹʆɻʍʏʉR ɶɿɲʏɻʆʉʋʉʀɲɿʍʖʑɸɿ
xx f x x
x11
13
2
2
1 1++
++^ h ɶɿɲʃɳɽɸx 0> .
Ɂɲɴʌɸɽɸʀʏʉ lim f xx" 3-^ h.
ϱϳ Ȱʆ lim xP x
2 2x + =" 3+
^ hʃɲɿ lim x
xP2x 0 hn l=
"
^ hʆɲɴʌɸɽɸʀʉʏʑʋʉʎʏʉʐʋʉʄʐʘ-
ʆʑʅʉʐʃɲɽʙʎʃɲɿʉ R!l .ϱϴ Ȱʆ ɻ ʍʐʆɳʌʏɻʍɻ :f R R" ɸʀʆɲɿ ʋɸʌɿʏʏɼ ʃɲɿ ɿʍʖʑɸɿ
lim f x x x x2 1 3x
2+ - + + =" 3+
^_ h i ,ʆɲɴʌɸɽɸʀʏʉ lim f xx" 3-^ h.
ϱϵ Ȱʆ ɿʍʖʑɸɿ x f x g x f x g x22 2 H+ -^ ^^ ^ ^^h hh h hh ɶɿɲ ʃɳɽɸ dž R! ʆɲɴʌɸɽʉʑʆʏɲʊʌɿɲ lim f x
x" 3-^ hʃɲɿ limg x
x" 3-^ h.
28
ϲϬ Ȱʆɿʍʖʑɸɿ xf x
x x1 1
2Ghn-^ h
ɶɿɲʃɳɽɸdž R! ʆɲɴʌɸɽɸʀʏʉ lim f xx" 3+^ h.
ϲϭ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ Ĩ ʉʌɿʍʅɹʆɻ ʍʏʉ R ɶɿɲ ʏɻʆ ʉʋʉʀɲ ɿʍʖʑɸɿx f x x x1 3 2 G+ -^ ^h h ɶɿɲʃɳɽɸx 0> Ɂɲɴʌɸɽɸʀʏʉ lim f x
x" 3+^ h.
ϲϮ Ȱʆ ɿʍʖʑɸɿ f x e2 1x xH+^ ^h h ɶɿɲ ʃɳɽɸ dž R! ʆɲ ʐʋʉʄʉɶɿʍʏɸʀ ʏʉ
lim f xx" 3+^ hȾɲʏʊʋɿʆʆɲɴʌɸɽɸʀʏʉ lim f x
f xf x
f x2 3 5
2 1x 3
3 2
+ -- +
" 3+ ^ ^^ ^h hh h.
ϲϯ Ȱʆ ɿʍʖʑɸɿ f x x x 14 G vyo -^ h ɶɿɲ ʃɳɽɸ dž R! ʆɲ ʐʋʉʄʉɶɿʍʏɸʀ ʏʉ
ʊʌɿʉ lim f x f x f x2 3x 0
2 - + +"
^ ^ ^_ h h hi.ϲϰ Ȱʆɿʍʖʑɸɿ f x x x exHhn^ h ɶɿɲʃɳɽɸ dž R! ʆɲʐʋʉʄʉɶɿʍʏɸʀʏʉʊʌɿʉ
lim f x f x f x1x 0
2 + + -"
^ ^ ^_ h h hi.ϲϱ Ȱʆ lim f x
f xx x0
3ba=+-
-" ^ hh ʊʋʉʐ , R!a b ʆɲɴʌɸɽɸʀʏʉ lim f x
x x0"^ h.
ϲϲ Ȱʆɿʍʖʑɸɿ lim xf x x1
x 203
vyo-=-
"
^ ^h hʆɲɴʌɸɽɸʀʏʉ lim f x
x 0"^ h.
ϲϳ Ȱʆɿʍʖʑɸɿ lim xf x x 1
x 13
hn r=-
-" ^^ ^ hh h
ʆɲɴʌɸɽɸʀʏʉ lim f xx 1"^ h.
ϲϴ Ȱʆ lim xf x e2
5x
x x1 2
hn =+" 3+
+ -^ hʆɲʐʋʉʄʉɶɿʍʏɸʀʏʉ lim f x
x" 3+^ h.
ϲϵ Ȱʆ ɿʍʖʑɸɿ f x f x2= -^ ^h h ɶɿɲ ʃɳɽɸ dž R! ʃɲɿlim f x x x1 2x
3 3- + + =" 3+^_ h i ,ʆɲʐʋʉʄʉɶɿʍʏɸʀʏʉ lim f x
x" 3-^ h.
ϳϬ Ȱʆɿʍʖʑɸɿ f x xf x xx2 1
11- = -
+^ `h j ɶɿɲʃɳɽɸx 1R! - " ,ʆɲɴʌɸɽʉʑʆɲʆʐʋɳʌʖʉʐʆʏɲʊʌɿɲ lim f x
x" 3+^ hʃɲɿ lim f x
x 1"^ h.
ϳϭ Ȱʆ lim f xx X0
3=+"^ h ʆɲʐʋʉʄʉɶɿʍʏɸʀʏʉ lim ln x f x2 3
x xf x 2
0+ + -
"^ ^^ h hh6 @.
ϳϮ ȶʍʏʘʊʏɿ lim xx
f x2 02
x 2 hn r+ =-
" ^ ^h h ʅɸf x 0<^ h ɶɿɲʃɳɽɸdž R! Ɂɲʐʋʉ-
ʄʉɶɿʍʏɸʀʏʉ lim f xf x
f xf x
6 23 2
5 2 47 2 1
x 0 2
2
++
- + ++ + -
" ^ ^^ ^h hh h.
ϳϯ ȶʍʏʘʊʏɿ limx x
x xf x1 1
0x 2 2 hn
hn
++
=" ` ^j h ʅɸ f x 0>^ h ɶɿɲ ʃɳɽɸ dž R!
Ɂɲʐʋʉʄʉɶɿʍʏɸʀʏʉ lim f xf x f x
2 13 2
x 0
2
-+ -
" ^^ ^hh h.
29
ϳϰ Ɂɲɸʇɸʏɲʍɽʉʑʆʘʎʋʌʉʎʏɻʍʐʆɹʖɸɿɲʉɿʋɲʌɲʃɳʏʘʍʐʆɲʌʏɼʍɸɿʎ
ɲ
,
,
,
f xx xx x x
x
xx x x
0
1 0
0
>
<2 hn
=+-
=+
^ hZ
[
\
]]
]]
ɴ,,
,f x
x xx x
x x
1 13 1 22 2
< <
2 G
H
=+ -- -^ h
Z
[
\
]]
].
ɶ,,f x
x xx x1 1
1 1>
2
2
G=
--^ h ( ɷ ,
,f x x x
x xx
01 0
2
!vyo= +=
^ h * .
ɸ ,
,f x e
e x
x1
3 1 0
3 0x
x1
1
!= +-
=^ h * ʍʏ ,
,ln
lnf x x
xx
x
11 0
0 0>!=
+
=^ ^h h* .
ϳϱ Ɂɲɸʇɸʏɲʍɽʉʑʆʘʎʋʌʉʎʏɻʍʐʆɹʖɸɿɲʉɿʋɲʌɲʃɳʏʘʍʐʆɲʌʏɼʍɸɿʎ
ɲ ,,
f x e xx x1 0
0<x
1
Hvyo= +^ h ) .
ɴ ,,
lnf x x x xx
1 1 0 10 1
< !vyo= -=
^ ^h h* .
ɶ ,,
f x xx x x
x x1
5 6 2 11
>3
2
Gvyo r= -
- + -^ ^h h* .
ɷ
,
,
,
xfxx x
x
xx x
11 1
21 1
13 2 1
>
<2
=--
=
-+ -
^ hZ
[
\
]]]
]]
.
ϳϲ ȶʍʏʘ ɻ ʍʐʆɳʌʏɻʍɻ,
,,
f xx x x
x xx x x
1 34 2 3 4
3 4
<
>
2
2
G Gab aa b
=- ++ +
- + + -^ ^
^h h
h* Ɂɲ ɴʌɸ-
ɽʉʑʆʏɲ , R!a b ʙʍʏɸɻĨʆɲɸʀʆɲɿʍʐʆɸʖɼʎϳϳ Ɂɲɴʌɸɽʉʑʆʉɿʏɿʅɹʎʏʉʐ R!a ʙʍʏɸʆɲɸʀʆɲɿʍʐʆɸʖɼʎɻʍʐʆɳʌ-
ʏɻʍɻ ,,
f x xx x
xx
02 1 0
3
2
!hn hn a
a=
- =^ ^^h hh* .
ϳϴ Ɂɲɴʌɸɽʉʑʆʉɿʏɿʅɹʎʏʘʆ , R!a b ʙʍʏɸʆɲɸʀʆɲɿʍʐʆɸʖɼʎɻʍʐ-
ʆɳʌʏɻʍɻ ,,
f x x xx x
xe x
4 222
>x
2
2
Ga= + -
- + -
b a-^ h * .
30
ϳϵ Ɂɲɴʌɸɽʉʑʆʉɿʏɿʅɹʎʏʘʆ , R!a b ʙʍʏɸʆɲɸʀʆɲɿʍʐʆɸʖɼʎɻʍʐ-
ʆɳʌʏɻʍɻ
,
,
,
f xx
xx
x
xx x
2 0
21 0
0
>
<
hn ahn b
b a
= =+ -
^^^
hhhZ
[
\
]]]
]]
.
ϴϬ ȶʍʏʘ ɻ ʍʐʆɳʌʏɻʍɻ, ,
, ,f xx x xx x x
1 0 11 1 3
!!
a ba b
=+ ++ +^ ^h h h66 @) Ɂɲ ɴʌɸ-
ɽʉʑʆʏɲ , R!a b ʙʍʏɸɻĨʆɲɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉʋɸɷʀʉʉʌɿʍʅʉʑʏɻʎʃɲɿ lim limf x f x4 12 1
x x49
41
= +" "
^ ^h h .
ϴϭ ȴʀʆɸʏɲɿ ɻ ʍʐʆɳʌʏɻʍɻ ,
,f x x
x x x
x2 2
2
3
!a b
c= -
+ +
=^ h * Ɂɲ ɴʌɸɽʉʑʆ ʏɲ
, , R!a b c ʙʍʏɸɻĨʆɲɸʀʆɲɿʍʐʆɸʖɼʎ
ϴϮ ȴʀʆɸʏɲɿɻʍʐʆɳʌʏɻʍɻ,
,,
f xx xx x
x x x
2 3 11 1 1
3 5 1< <2
2
G
H
a b
b a=
+ -+ -+ -
^ h * Ɂɲɴʌɸɽʉʑʆ
ʏɲ , R!a b ʙʍʏɸɻĨʆɲɸʀʆɲɿɲ ʍʐʆɸʖɼʎɴ ʍʐʆɸʖɼʎʍʏʉx 10 = ʃɲɿɲʍʐʆɸʖɼʎʍʏʉʎx 10 =-
ϴϯ Ȱʆ x x f x x x2 1 13 vyo- = - -^ ^ ^ ^h h h h ɶɿɲʃɳɽɸ x 0H ʆɲɴʌɸɽɸʀʉʏʑʋʉʎʏɻʎʍʐʆɸʖʉʑʎʍʐʆɳʌʏɻʍɻʎĨʍʏʉ ,0 3+ h6 .
ϴϰ ɁɲɴʌɸɽɸʀʉʏʑʋʉʎʏɻʎʍʐʆɸʖʉʑʎʍʐʆɳʌʏɻʍɻʎĨɶɿɲʏɻʆʉʋʉʀɲɿʍʖʑ-ɸɿxf x x x x3 2hn= + -^ h ɶɿɲʃɳɽɸʋʌɲɶʅɲʏɿʃʊɲʌɿɽʅʊdž
ϴϱ ɁɲɴʌɸɽɸʀʉʏʑʋʉʎʏɻʎʍʐʆɸʖʉʑʎʍʐʆɳʌʏɻʍɻʎĨɶɿɲʏɻʆʉʋʉʀɲɿʍʖʑ-ɸɿxf x x x x5 2 2hn= + -^ h ɶɿɲʃɳɽɸʋʌɲɶʅɲʏɿʃʊɲʌɿɽʅʊdž
ϴϲ Ȱʆ x x xf x1 1 1G Ghn ` j ɶɿɲʃɳɽɸ x 0! ʆɲɸʇɸʏɲʍʏɸʀʘʎʋʌʉʎ ʏɻ
ʍʐʆɹʖɸɿɲʍʏʉx 00 = ɻʍʐʆɳʌʏɻʍɻ ,,
g x xf x
xx
01 1
!==
^ ^h h* .
ϴϳ Ȱʆf x xf x22 G^ ^h hɶɿɲʃɳɽɸdž R! ʆɲɷɸɿʖɽɸʀʊʏɿɲ ɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉϬ
ɴ lim xf x 0
x 2 =" 3-
^ h.
ϴϴ Ȱʆ ɶɿɲ ʏɻ ʍʐʆɳʌʏɻʍɻ Ĩ ʉʌɿʍʅɹʆɻ ʍʏʉ R ɶɿɲ ʃɳɽɸ ,x 0! d6 @ɿʍʖʑɸɿ x x f x x4
8 1 1G Ghn
a+ + + + +^ h ʆɲ ɴʌɸɽɸʀ ɻ
31
ʏɿʅɼ ʏʉʐ R!a ʙʍʏɸ ʆɲ ɸʀʆɲɿ ʍʐʆɸʖɼʎ ʍʏʉ Ϭ ɻ ʍʐʆɳʌʏɻʍɻ
,,
g x xf x f
xx f x
00
4 0 0>
2 Ga=
-
+ -^ ^ ^
^h h hh* .
ϴϵ Ȱʆ f x f y x yG- -^ ^h h ɶɿɲʃɳɽɸ ,x y R! ʆɲɷɸʀʇɸʏɸʊʏɿɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉR .
ϵϬ Ȱʆ lim xx f x x
11
2x 1
hn r-
- -=
"
^ ^ ^h h hʃɲɿɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎ
ʆɲɴʌɸɽɸʀʏʉf 1^ h.ϵϭ Ȱʆ f x x2Hvyo r-^ h ɶɿɲʃɳɽɸ ,x 0! r6 @ʊʋʉʐĨʍʐʆɳʌʏɻʍɻʍʐʆɸ-
ʖɼʎʍʏʉR ʆɲɴʌɸɽɸʀʏʉf 2r` j.
ϵϮ Ȱʆ f 0 4=^ h ʃɲɿ f x x x f x1 22 2hn+ = +^ ^h h ɶɿɲ ʃɳɽɸ x R*! ʆɲɷɸɿʖɽɸʀʊʏɿɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉϬ
ϵϯ ȰʆɶɿɲʏɻʍʐʆɳʌʏɻʍɻĨɿʍʖʑɸɿ f x x f x3 2 hn= +^ ^h hɶɿɲʃɳɽɸdž R! ʆɲɷɸʀʇɸʏɸʊʏɿɲ f x xG^ h ɶɿɲʃɳɽɸdž R! .ɴ ɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉx 00 = .
ϵϰ ȰʆɶɿɲʏɻʍʐʆɳʌʏɻʍɻĨɿʍʖʑɸɿ xf x x f x x x1 2 12Ghn- - - + +^ ^ ^h h h ɶɿɲʃɳɽɸdž R! ʃɲɿɸʀʆɲɿʍʐʆɸʖɼʎʆɲɸʇɸʏɳʍɸʏɸɲʆɻɶʌɲʔɿʃɼʋɲ-ʌɳʍʏɲʍɻʏɻʎĨʋɸʌʆɳɲʋʊʏʉʍɻʅɸʀʉ ,A 1 1^ h.
ϵϱ Ȱʆ ɻ ʍʐʆɳʌʏɻʍɻ Ĩ ɸʀʆɲɿ ʍʐʆɸʖɼʎ ʍʏʉ x 00 = ʃɲɿ ɿʍʖʑɸɿxf x x1 5< vyo+^ h ɶɿɲʃɳɽɸx R*! ʆɲɴʌɸɽɸʀʏʉf 0^ h.
ϵϲ ȰʆɶɿɲʏɻʍʐʆɳʌʏɻʍɻĨɿʍʖʑɸɿf x f x2 - =^ ^h hɶɿɲʃɳɽɸdž R! ʃɲɿɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉϯʆɲɷɸʀʇɸʏɸʊʏɿɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉͲϭ
ϵϳ Ȱʆ ɻ ʍʐʆɳʌʏɻʍɻ Ĩ ɸʀʆɲɿ ʍʐʆɸʖɼʎ ʍʏʉ Ϭ ʃɲɿ ɿʍʖʑɸɿf x f x x x1 12+ + = + +^ ^h h ɶɿɲʃɳɽɸ dž R! ʆɲɷɸʀʇɸʏɸʊʏɿɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉϭ
ϵϴ ȶʍʏʘʊʏɿf x y f x f y+ =^ ^ ^h h hɶɿɲʃɳɽɸ ,x y R! ȰʆɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉϬʆɲɷɸɿʖɽɸʀʊʏɿɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉR .
ϵϵ ȶʍʏʘʊʏɿf x y e f y e f x xyx y+ = + +^ ^ ^h h h ɶɿɲʃɳɽɸ ,x y R! ȰʆɻĨɸʀ-ʆɲɿʍʐʆɸʖɼʎʍʏʉϬʆɲɷɸɿʖɽɸʀʊʏɿɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉR .
ϭϬϬ Ɂɲɷɸɿʖɽɸʀʊʏɿʉɿʋɲʌɲʃɳʏʘɸʇɿʍʙʍɸɿʎɹʖʐʆʏʉʐʄɳʖɿʍʏʉʆʅʀɲʌʀɺɲʍʏʉɲʆʏʀʍʏʉɿʖʉɷɿɳʍʏɻʅɲɲ x x2 3 0hn vyo+ = ʍʏʉ ,0 r^ h.ɴ x x 13 = + ʍʏʉ ,1 2^ h.ɶ x x 4vyo+ = ʍʏʉ ,2r r^ h.ɷ x
xxx1 1 0
2 6
a b-+ + -
+ = ʍʏʉ ,a b^ h.
32
ɸ xx
xx
11
23 0
10 8
-+ + -
+ = ʍʏʉ ,1 2^ h.ʍʏ x
xxx
4 3 0rf
rv
- + - = ʍʏʉ ,4 3r r` j.
ϭϬϭ ȶʍʏʘ ʍʐʆɲʌʏɼʍɸɿʎ Ĩ Ő ʍʐʆɸʖɸʀʎ ʍʏʉ ,a b6 @ ʃɲɿf gb a=^ ^h h f g g!a b a=^ ^ ^h h h.Ɂɲɷɸʀʇɸʏɸʊʏɿʐʋɳʌʖɸɿʏʉʐʄɳʖɿ-ʍʏʉʆɹʆɲ ,!t a b^ hf gt t=^ ^h h.
ϭϬϮ ȶʍʏʘĨ ŐʍʐʆɸʖɸʀʎʍʏʉR ʃɲɿ f x g x0G G^ ^h hɶɿɲʃɳɽɸdž R! ʃɲɿʐʋɳʌʖʉʐʆ , R!a b ʙʍʏɸ f a a=^ h ʃɲɿ g b b=^ h Ɂɲ ɷɸʀʇɸʏɸ ʊʏɿR!c ʏɹʏʉɿʉʙʍʏɸf gc c c+ =^ ^h h .
ϭϬϯ ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,0 16 @ʃɲɿf 0 a=^ h ʃɲɿf 1 b=^ h ʅɸ , ,0 1!a b ^ hɁɲɷɸɿʖɽɸʀʊʏɿɻɸʇʀʍʘʍɻ f x x=^ h ɹʖɸɿ ʏʉʐʄɳʖɿ-ʍʏʉʆʅɿɲʌʀɺɲʍʏʉ ,0 1^ h.
ϭϬϰ Ȱʆɻʍʐʆɳʌʏɻʍɻ : , ,f "a a a a- -6 6@ @ 0>a ɸʀʆɲɿɶʆɻʍʀʘʎɲʑʇʉʐʍɲʃɲɿʍʐʆɸʖɼʎʆɲɷɸʀʇɸʏɸʊʏɿɻCf ɹʖɸɿɹʆɲʅʊʆʉʃʉɿʆʊʍɻʅɸʀʉʅɸʏɻʆy x=- .
ϭϬϱ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ : ,f R"a b6 @ ʍʐʆɸʖɼʎ ʅɸ f 2a b=_ i ʃɲɿf 2b a=^ h .Ɂɲɲʋʉɷɸʀʇɸʏɸʊʏɿ ʐʋɳʌʖɸɿ ,!i a b^ hʙʍʏɸ f 2i i=^ h ʊʋʉʐ , R!a b ʅɸ <a b.
ϭϬϲ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ : , ,f 0 1 0 1"7 7A A ʍʐʆɸʖɼʎ Ɂɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿʐʋɳʌʖɸɿ ,e1!i 7 Aʙʍʏɸ ln lnf i i=^ h .
ϭϬϳ ȶʍʏʘ , : , ,f g "a b a b6 6@ @ ʍʐʆɸʖɸʀʎ ʍʐʆɲʌʏɼʍɸɿʎ Ȱʆ f a a=_ i ʃɲɿg b b=^ h , ʆɲɲʋʉɷɸʀʇɸʏɸʊʏɿ ʐʋɳʌʖɸɿ ʏʉʐʄɳʖɿʍʏʉʆ ɹʆɲ ,!i a b6 @ʙʍʏɸ f g$ $ $l i m i l m i+ = +^ ^ ^h h h ʊʋʉʐ , R!l m ʉʅʊʍɻʅʉɿ ʃɲɿ, R!a b ʅɸɲфɴ
ϭϬϴ ȰʆɻʍʐʆɳʌʏɻʍɻĨɸʀʆɲɿʍʐʆɸʖɼʎʍʏʉ ,0 16 @ʃɲɿ f x1 0< <- ^ h ʆɲɷɸɿ-ʖɽɸʀʊʏɿɻɸʇʀʍʘʍɻ f x f x x2 + =-^ ^h h ɹʖɸɿʅʀɲʏʉʐʄɳʖɿʍʏʉʆʌʀɺɲʍʏʉ,0 1^ h.
ϭϬϵ Ȱʆ : ,f 0 2 R"a6 @ ʍʐʆɸʖɼʎʅʎ f f0 2a=^ ^h hʆɲɷɸɿʖɽɸʀʊʏɿʐʋɳʌʖɸɿ,0!p a^ hʙʍʏɸf fp p a= +^ ^h h.
ϭϭϬ ȶʍʏʘʍʐʆɳʌʏɻʍɻ : ,f R"a b6 @ ʊʋʉʐ , R!a b ʅɸɲфɴɁɲɲʋʉɷɸʀ-ʇɸʏɸʊʏɿ
ɲ Ȱʆɲфʅфʆфɴʏʊʏɸʐʋɳʌʖɸɿ ,x0 ! a b^ hʙʍʏɸ f ff x2 0
n o+=
` _ _j i i.ɴ Ȱʆɲфʅфʆфɴʃɲɿ ,1 2i i ɽɸʏɿʃʉʀɲʌɿɽʅʉʀɲʃɹʌɲɿʉɿʏʊʏɸʐʋɳʌ-
ʖɸɿ ,x0 ! a b^ hʙʍʏɸf xf f
01 2
1 2
i ii n i o
= ++_ ` _i j i
.
33
ϭϭϭ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ : ,f R"a b6 @ ʍʐʆɸʖɼʎ ȴʀʆɸʏɲɿ ʊʏɿ
f f 32
aa b= +_ ci m Ɂɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ʐʋɳʌʖɸɿ ,!i a bh6 ʙʍʏɸ
f f 3i ib a= + -^ ch m,ʊʋʉʐ , R!a b ʅɸɲфɴ
ϭϭϮ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ Ĩ ɶʆɻʍʀʘʎ ɲʑʇʉʐʍɲ ʃɲɿ ʍʐʆɸʖɼʎ ʍʏʉ ,a b6 @ʃɲɿ ʍʐʆɳʌʏɻʍɻ Ő ɶʆɻʍʀʘʎʔɽʀʆʉʐʍɲ ʃɲɿ ʍʐʆɸʖɼʎ ʍʏʉ ,a b6 @ Ȱʆ
, ,f g+ 4!a b a b^ ^h h6 6@ @ , ʆɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ʐʋɳʌʖɸɿ ʅʉʆɲɷɿʃʊ,!i a b6 @ʙʍʏɸf gi i=^ ^h h.
ϭϭϯ ȴʀʆɸʏɲɿɻʍʐʆɳʌʏɻʍɻĨʅɸf x x x x3 2a b ab ab= - + - +_ ^i h ,
ʊʋʉʐ , R!a b Ɂɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ʐʋɳʌʖɸɿ ,0 1!i 7 A ʙʍʏɸf 2
1 1i
a b= - -^ ^ ^h h h.
ϭϭϰ Ɂɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ɻ ɸʇʀʍʘʍɻ x x16 116 - = ɹʖɸɿ ʏʉʐʄɳʖɿʍʏʉ ʅɿɲʌʀɺɲʍʏʉR .
ϭϭϱ ȰʆɲɴхϬʆɲɲʋʉɷɸʀʇɸʏɸʊʏɿɻɸʇʀʍʘʍɻ x x2 2avyo b+ = ɹʖɸɿʅɿɲʏʉʐʄɳʖɿʍʏʉʆɽɸʏɿʃɼʋʌɲɶʅɲʏɿʃɼʄʑʍɻʋʉʐɷɸʆɸʀʆɲɿʅɸɶɲʄʑʏɸʌɻʏʉʐɲнɴ
ϭϭϲ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ :f RR " ʍʐʆɸʖɼʎ ʅɸ f x x x3 2a b= + +_ i ʊʋʉʐ0>b ʃɲɿɲнɴф ͲϭɁɲɲʋʉɷɸʀʇɸʏɸʊʏɿɻɸʇʀʍʘʍɻ f x 0=_ i ɹʖɸɿ
ʏʌɸɿʎʌʀɺɸʎʋʌɲɶʅɲʏɿʃɹʎϭϭϳ ȶʍʏʘ , :f g RR " ɷʑʉʍʐʆɸʖɸʀʎʍʐʆɲʌʏɼʍɸɿʎʙʍʏɸf x g x xl- =_ _i i
ɶɿɲʃɳɽɸdž R! ʊʋʉʐʃʍʏɲɽɸʌʊʎʋʌɲɶʅɲʏɿʃʊʎɲʌɿɽʅʊʎȰʆɻɶʌɲ-ʔɿʃɼʋɲʌɳʍʏɲʍɻʏɻʎŐʏɹʅʆɸɿʏʉʆɳʇʉʆɲ 'x x ʍɸɷʑʉʍɻʅɸʀɲʅɸɸʏɸ-ʌʊʍɻʅɸʎʏɸʏʅɻʅɹʆɸʎʆɲɲʋʉɷɸʀʇɸʏɸʊʏɿɻɸʇʀʍʘʍɻ f x 0=_ i ɹʖɸɿʅɿɲʏʉʐʄɳʖɿʍʏʉʆʌʀɺɲʍʏʉR .
ϭϭϴ ȶʍʏʘʍʐʆɳʌʏɻʍɻĨʍʐʆɸʖɼʎʍʏʉR ʃɲɿɷɿɳʍʏɻʅɲ , R3a b6 @ ȰʆɻĨɿʃɲʆʉʋʉɿɸʀʏɿʎʋʌʉʒʋʉɽɹʍɸɿʎʏʉʐɽɸʘʌɼʅɲʏʉʎŽůnjĂŶŽʍʏʉ ,a b6 @ʆɲɲʋʉɷɸʀʇɸʏɸʊʏɿɻɸʇʀʍʘʍɻ f f x f x f 02a l b+ + =_ _ _ ^i i i h ʊʋʉʐʃʋʌɲɶʅɲʏɿʃʊʎɲʌɿɽʅʊʎɹʖɸɿɷʑʉʄʑʍɸɿʎʍʏʉR .
ϭϭϵ ȶʍʏʘ ʉɿ ʍʐʆɸʖɸʀʎ ʍʐʆɲʌʏɼʍɸɿʎ , : ,f g R"a b6 @ ʊʋʉʐ , R!a b ʅɸ ɲ ф ɴ ȶʍʏʘ ɸʋʀʍɻʎ ʊʏɿ , ,f f ga b b a b b= = =_ ^ ^i h h ʃɲɿg a a=_ i .Ɂɲɲʋʉɷɸʀʇɸʏɸʊʏɿʐʋɳʌʖɸɿʏʉʐʄɳʖɿʍʏʉʆɹʆɲ ,!i a b^ hʙʍʏɸ f g g f 2% %i i i+ =_ ^ _ ^i h i h .
34
ϭϮϬ ȶʍʏʘ ʍʐʆɲʌʏɼʍɸɿʎ Ĩ Ő ʍʐʆɸʖɸʀʎ ʍʏʉ ɷɿɳʍʏɻʅɲ ,a a-8 B ʊʋʉʐR*!a ʙʍʏɸ ɻ Ĩ ʆɲ ɸʀʆɲɿ ʋɸʌɿʏʏɼ ɻ Őʔɽʀʆʉʐʍɲ ʃɲɿ ʆɲ ɿʍʖʑɸɿ
g a a=-_ i ʃɲɿg a a- =_ i .Ɂɲɲʋʉɷɸʀʇɸʏɸʊʏɿʐʋɳʌʖɸɿʏʉʐʄɳʖɿʍʏʉʆ
ɹʆɲ ,x0 ! a a-_ iʙʍʏɸʆɲɿʍʖʑɸɿf g x f x g x 00 0 0+ + =__ _ _ii i i .ϭϮϭ ȶʍʏʘ ʍʐʆɲʌʏɼʍɸɿʎ , : ,f g R"a b6 @ ʍʐʆɸʖɸʀʎ ʙʍʏɸ
, , ,f ga b a b a b= =^ ^h h6 6 6@ @ @.Ɂɲɲʋʉɷɸʀʇɸʏɸʊʏɿʐʋɳʌʖɸɿ ,!i a b6 @ʙʍʏɸʆɲɿʍʖʑɸɿf g g f 2i i i+ =^^ ^^hh hh .
ϭϮϮ ȴʀʆʉʆʏɲɿ ʉɿ ʍʐʆɸʖɸʀʎ ʍʐʆɲʌʏɼʍɸɿʎ, , : , ,f g h "a b a b6 6@ @ ʊʋʉʐ , R!a b ʅɸ ɲ ф ɴ ʙʍʏɸ ɻ Ĩ ʆɲɸʀʆɲɿ ɶʆɻʍʀʘʎ ɲʑʇʉʐʍɲ ɸʆʙ ɻ Ő ɶʆɻʍʀʘʎ ʔɽʀʆʉʐʍɲ ʃɲɿ
, , ,f ga b a b a b= =^ ^h h6 6 6@ @ @. ȶʍʏʘ ɸʋʀʍɻʎ ʊʏɿ h a b=_ i ʃɲɿh b a=^ h . Ɂɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ʐʋɳʌʖɸɿ ,!i a b^ h ʙʍʏɸ ʆɲ ɿʍʖʑɸɿf g h g fi i=^^^ ^^hhh hh.
ϭϮϯ ȶʍʏʘʍʐʆɳʌʏɻʍɻĨʍʐʆɸʖɼʎʙʍʏɸx x xf x x x2 8 4 6
3 2 6G G hn+ - + +_ i ;ϭͿ
ɶɿɲʃɳɽɸ ,x 4 3! - + i7 .
ɲ Ɂɲɲʋʉɷɸʀʇɸʏɸʊʏɿf 0 61=^ h .
ɴ Ɂɲɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ʐʋɳʌʖɸɿ ʏʉʐʄɳʖɿʍʏʉʆ ɹʆɲ ,0 1!l _ A ʙʍʏɸf 6
6l hn l l= +_ i .ϭϮϰ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ Ĩ ʍʐʆɸʖɼʎ ʍʏʉ R ʙʍʏɸ ʆɲ ɿʃɲʆʉʋʉɿɸʀ ʏɿʎ ʍʐʆ-
ɽɼʃɸʎ lim xf x
1 4x 1 - ="
_ i ʃɲɿ x x f x x4 2 2 42G Ghn - - -^ ^ _h h i ɶɿɲ ʃɳɽɸ
dž R! .Ɂɲɲʋʉɷɸʀʇɸʏɸʊʏɿɻɶʌɲʔɿʃɼʋɲʌɳʍʏɲʍɻcf ʏɻʎʍʐʆɳʌʏɻ-ʍɻʎĨʏɹʅʆɸɿʏɻʆɶʌɲʔɿʃɼʋɲʌɳʍʏɲʍɻʏɻʎʋɲʌɲɴʉʄɼʎy x x 12= - + ʍɸʍɻʅɸʀʉʅɸʏɸʏʅɻʅɹʆɻʍʏʉɷɿɳʍʏɻʅɲ ,1 2_ i.
ϭϮϱ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ Ĩ ʍʐʆɸʖɼʎ ʍʏʉ ,0 16 @ ʃɲɿ ɶʆɻʍʀʘʎ ɲʑ-ʇʉʐʍɲ ʅɸ f 0 2=^ h ʃɲɿ f 1 4=^ h Ɂɲ ɷɸɿʖɽɸʀ ʊʏɿ ɻ ɸʇʀʍʘʍɻ
xff f f f
451
52
53
54
=+ + +^ ` ` ` `h j j j j
ɹʖɸɿʅʀɲʏʉʐʄɳʖɿʍʏʉʆʄʑʍɻʍʏʉ,0 1^ h.
ϭϮϲ ȶʍʏʘʍʐʆɳʌʏɻʍɻĨʍʐʆɸʖɼʎʍʏʉɷɿɳʍʏɻʅɲ ,a b6 @ʊʋʉʐ , R!a b ʅɸ<a b ʃɲɿ , , , , ,x x x x x1 2 3 4 5 ! a b6 @. Ɂɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ʐʋɳʌʖɸɿ
,!l a b^ hʙʍʏɸ f f x f x f x f x f x15 2 3 4 51 2 3 4 5l = + + + +_ _ _ _ _ _i i i i i i.
35
ϭϮϳ Ȱʆ ɻ ʍʐʆɳʌʏɻʍɻ Ĩ ɸʀʆɲɿ ʍʐʆɸʖɼʎ ʍʏʉ ɷɿɳʍʏɻʅɲ ,a b6 @ ʊʋʉʐ, R!a b ʅɸ ɲ ф ɴ ʆɲ ɲʋʉɷɸʀʇɸʏɸ ʊʏɿ ʐʋɳʌʖɸɿ ,!i a b6 @ ʙʍʏɸ
f f f f91 2 3 2 4i a
a bb= + + +^ _ c ^h i m h< F.
ϭϮϴ ȶʍʏʘʍʐʆɲʌʏɼʍɸɿʎĨ Őʍʐʆɸʖɸʀʎʍʏʉɷɿɳʍʏɻʅɲ ,a b6 @ʙʍʏɸʆɲɿʍʖʑɸɿg x 0!_ i ɶɿɲʃɳɽɸ ,x ! a b6 @Ɂɲɲʋʉɷɸʀʇɸʏɸʊʏɿʐʋɳʌʖɸɿ ,!p a b^ hʏɹʏʉɿʉʙʍʏɸ g
f 1 1pp
p a p b= - + -^ hh .
ϭϮϵ ȶʍʏʘʍʐʆɳʌʏɻʍɻĨʍʐʆɸʖɼʎʍʏʉɷɿɳʍʏɻʅɲ ,a b6 @ʃɲɿ , ,1 2 !t t a b^ hɷʑʉɷɿɲɷʉʖɿʃɹʎʄʑʍɸɿʎʏɻʎɸʇʀʍʘʍɻʎ f x 0=_ i Ɂɲɲʋʉɷɸʀʇɸʏɸʊʏɿʍʏʉɷɿɳʍʏɻʅɲ ,1 2t t` jɻʍʐʆɳʌʏɻʍɻĨɷɿɲʏɻʌɸʀʍʏɲɽɸʌʊʋʌʊʍɻʅʉ
ϭϯϬ Ȱʆɶɿɲʃɳɽɸ ,x 1 0! -6 @ɿʍʖʑɸɿʊʏɿx xf x2 83 2+ =^ h ʃɲɿɻĨʍʐʆɸʖɼʎʆɲɷɸɿʖɽɸʀʊʏɿɻĨɷɿɲʏɻʌɸʀʍʏɲɽɸʌʊʋʌʊʍɻʅʉ
ϭϯϭ Ȱʆɶɿɲʃɳɽɸ ,x ! a b6 @ɿʍʖʑɸɿʊʏɿ f x 0!^ h ʃɲɿɻĨʍʐʆɸʖɼʎʆɲɷɸɿ-ʖɽɸʀʊʏɿɻĨɷɿɲʏɻʌɸʀʍʏɲɽɸʌʊʋʌʊʍɻʅʉ
ϭϯϮ ȶʍʏʘʍʐʆɳʌʏɻʍɻĨʍʐʆɸʖɼʎʍʏʉR ʃɲɿɲхϬȺɸʘʌʉʑʅɸʏʉʆʅɿɶɲɷɿʃʊ
ɲʌɿɽʅʊz x if x= + _ iʊʋʉʐdž R! Ȱʆɸʀʆɲɿ ,f f2 2 23
2a a a a= =-c cm m
ʃɲɿ ɶɿɲ ʃɳʋʉɿʉ ,x 0 20 ! a_ i ɿʍʖʑɸɿ ,z z2< Ha a a a- - ʃɲɿz2 3 Ha a- ,ʆɲɲʋʉɷɸʀʇɸʏɸʊʏɿf 0a =_ i .
ϭϯϯ Ɂɲɴʌɸɽɸʀʏʉʍʑʆʉʄʉʏɿʅʙʆʏʘʆʋɲʌɲʃɳʏʘʍʐʆɲʌʏɼʍɸʘʆɲ f x x x 4vyo= - -^ h ,x 2! r r^ h.ɴ xf e 1x= -^ h ,x 0 3! + h6 .ɶ lnf x e x1 1x= - + +^ ^h h ,x 0 3! + h6 .ɷ lnf x x x=^ h ,x 0 3! +^ h.
ϭϯϰ ȶʍʏʘ ʍʐʆɳʌʏɻʍɻ :f R R" ʍʐʆɸʖɼʎ ʅɸ f 0 0=^ h ʃɲɿlim f xx
3=+" 3-^ h Ɂɲ ɷɸʀʇɸʏɸ ʊʏɿ ʐʋɳʌʖɸɿ ʏʉʐʄɳʖɿʍʏʉʆ ɹʆɲ x 0<0
ʙʍʏɸf x e x x1x
0 00
0 hn= +^ h .
ϭϯϱ ȴʀʆɸʏɲɿɻʍʐʆɳʌʏɻʍɻ : ,f 0 R"3+^ h ʅɸf x x x1 1= - +^ h .ɲ ɁɲɴʌɸɽɸʀʏʉʍʑʆʉʄʉʏɿʅʙʆʏɻʎĨ .ɴ ɁɲɷɸʀʇɸʏɸʊʏɿɻĨɸʀʆɲɿɲʆʏɿʍʏʌɹʗɿʅɻʃɲɿʊʏɿɻf 1- ɸʀʆɲɿɶʆɻʍʀʘʎ
ʔɽʀʆʉʐʍɲɶ Ȱʆ ɽɸʘʌɼʍʉʐʅɸ ʊʏɿ ɻ f 1- ɸʀʆɲɿ ʍʐʆɸʖɼʎ ʆɲ ɴʌɸɽɸʀ ʏʉ
lim x f xf x x
x 1
1
+-
" 3+ -
-
^^ hh .