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 .