of 13
8/12/2019 Limit of Function Theory_e
1/13
FI^GZ
"filosg`ufirpgysojs.ol" 9
Neholotocl 1 Dofot ch i huljtocl h(x) os sion tc exost, is x i wgel,;g
ofh (i g) 6 ;g
of h (i + g) 6 Holote
(Deht giln dofot) (]obgt giln d ofot)
Lcte tgit we ire lct olteresten ol `lcwolb imcut wgit gippels it x 6 i. Idsc lcte tgit oh D.G.D. &
].G.D. ire mctg telnolb tcwirns ' ' cr , tgel ot os sion tc me olholote dofot.]efefmer, xi feils tgit x os ipprcijgolb tc i mut lct equid tc i.
Hulnifelti d tgecrefs cl d ofots1Det ix
of h (x) 6 iln ix
of b (x) 6 f. Oh & f ire holote, tgel1
(I) ixof { h (x) b (x) } 6 f
(M) ixof { h(x). b(x) } 6 . f
(J) ixof h x
b x
( )
( )6
f
, prcvonen f;
(N) ixof ` h(x) 6 ` ix
of h(x) 6 `0 wgere ` os i jclstilt.
(E) ixof h (b(x )) 6 h
)x(bof
ix
6 h (f)0 prcvonen h os jcltolucus it b (x) 6 f.
Exifpde # 9 1 Eviduite tge hcddcwolb dofots 1 -
(o)=x
of (x + =) (oo)
;xof jcs (sol x)
Zcdutocl 1 (o) x + = meolb i pcdylcfoid ol x, ots dofot is x = os bovel my=x
of (x + =) 6 = + = 6 >
(oo);x
of jcs (sol x) 6 jcs
xsolof
;x 6 jcs ; 6 9
Zedh prijtoje prcmdefs
Eviduite tge hcddcwolb dofots 1 -
(9)=x
of x(x 9) (=)
=xof
=x
>x=
Ilswers 1 (9) = (=) =
Dofot ch Huljtocl
8/12/2019 Limit of Function Theory_e
2/13
FI^GZ
"filosg`ufirpgysojs.ol" =
Olneterfolite hcrfs 1Oh cl puttolb x 6 i ol h(x), ily cle ch
;
;,
,; , , ;,9hcrf os cmtiolen, tgel tge dofotgis il olneterfolite hcrf. Idd tge imcve hcrfs ire olterjgilbeimde, o.e. we jil jgilbe cle hcrf tc
ctger my suotimde sumstotutocls etj.
Ol sujg jisesix
of h(x) fiy exost.
Jclsoner h(x) 6=x
>x=
. Gere=x
of x= > 6 ; iln
=xof x = 6 ;
=x
of h(x) gis il olneterfolite hcrf ch tge type
;
;.
xof
x
xlgis il olneterfolite hcrf ch type
.
;xof (9 + x)9/x os il olneterfolite hcrf ch tge type 9
LC^E 1
(o) + 6(oo) x 6
(ooo)i
6 ;, oh i os holote.
(ov)i
; os lct neholen hcr ily i].
(v) ;x of
x
x
os il olneterfolite hcrf wgereis ;x of
=
=
x
VxP
os lct il olneterfolite hcrf (wgere P.V
represelts breitest olteber huljtocl)
Ztunelts fiy refef mer tgese hcrf s idclbwotg tge prehox telnolb tc
o.e.zerctctelnolb
zerctctelnolbos il olneterfolite hcrf wgereis
zerctctelnolb
zercexijtdyos lct il olneterfolite
hcrf, ots vidue os zerc.
sofodirdy (telnolb tc cle)telnolb tc os olneterfolite hcrf wgereis (exijtdy cle)telnolb tc os lct il
olneterfolite hcrf, ots vidue os cle.
^c eviduite i dofot, we fust idwiys put tge vidue wgere ' x ' os ipprcijgolb tc ol tge huljtocl. Oh we bet
i neterfolite hcrf, tgel tgit vidue mejcfes tge dofot ctgerwose oh il olneterfolite hcrf jcfes, we
give tc refcve tge olneterfoliljy, clje tge olneterfoliljy os refcven tge dofot jil me eviduiten my
puttolb tge vidue ch x, wgere ot os ipprcijgolb.
Fetgcns ch refcv olb olneterfoliljyMisoj fetgcns ch refcvolb olneterfoliljy ire
(I) Hijtcrositocl (M) ]itoclidositocl
(J) U solb s tilnirn d ofots (N) Zums to tutocl
(E) Expilsocl ch huljtocls.
8/12/2019 Limit of Function Theory_e
3/13
FI^GZ
"filosg`ufirpgysojs.ol" :
Hijtcro si tocl fetgcnRe jil jiljed cut tge hijtcrs wgojg ire deinolb tc olneterfoliljy iln holn tge dofot ch tge refiololb
expressocl.
Exifpde # = 1 (o):x
of
:x>x
:x=x=
=
(oo)
=xof
x=x:x
):x=(=
=x
9=:
Zcdutocl 1 (o):x
of
:x>x
:x=x=
=
6:x
of
)9x)(:x(
)9x)(:x(
6 =
(oo)=x
of
x=x:x
):x=(=
=x
9=: 6 =x
of
)=x)(9x(x
):x=(=
=x
9
6=x
of
)=x)(9x(x
):x=(=)9x(x
6 =x of
)=x)(9x(x
8x5x=
6=x
of
)=x)(9x(x
):x)(=x(6
=xof
)9x(x
:x6
=
9
]itoclid osit ocl fetgcnRe jil ritoclidose tge orritoclid expressocl ol luferitcr cr nelcfolitcr cr ol mctg tc refcve tge
olneterfoliljy.
Exifpde # : 1 Eviduite 1
(o)9x
of
9x:=
9x95>
(oo)
;xof
x
x9x9
Zcdutocl 1 (o)9x
of
9x:=
9x95>
69x
of
)9x:=)(9x95>)(9x:=(
)9x95>)(9x:=)(9x95>(
69x
of
)x::(
)x9595(
9x95>
9x:=
6
=
5
(oo) ^ge hcrf ch tge bovel dofot os;
;wgel x;. ]itoclidosolb tge luferitcr, we bet
;xof
x
x9x9 6
;xof
x9x9
x9x9
x
x9x9
6 ;xof
)x9x9(x
)x9()x9(6 ;xof
)x9x9(x
x=
8/12/2019 Limit of Function Theory_e
4/13
FI^GZ
"filosg`ufirpgysojs.ol" >
6;x
of
x9x9
=6
=
=6 9
Zedh prijtoje prcmdefs
Eviduite tge hcddcwolb dofots 1 -
(:)9x
of
:xx=
9x):x=(=
(>)
=x
of
:/=
:/9
)x(sol9
)x(sol9
(5);g
of
g
xgx (8)
ixof
== ix
mimx
(2) ;xof
xx>
x
Ilswers 1 (:) 9;
9
(>) =
9
(5) x=
9
(8) mii>
9
(2) ;
Ztilnirn d ofots 1(i) (o)
;xof
x
xsol6
;xof
x
xtil6 9 P R gere x os feisuren ol rinoils V
(oo);x
of
x
xtil 96 ;x
of
x
xsol 96 9
(ooo);x
of
x9
)x9( 6 e 0 ix9
;x e)ix9(
of
(ov) xof
x
x
99
6 e 0 i
x
x ex
i9of
(v) ;xof
x
9ex 6 9 0 ;x
of
x
9ix 6 dcb
ei 6li ,i < ;
(vo) ;xof
x
)x9(l 6 9
(voo) ix of ixix
ll
6 lil 9
(m) Oh h(x) ;, wgel x i, tgel
(o)ix
of
)x(h
)x(hsol6 9 (oo)
ixof jcs h(x) 6 9
(ooo)ix
of
)x(h
)x(htil6 9 (ov)
ixof
)x(h
9e )x(h 6 9
(v)ix
of
)x(h9m )x(h 6l m, (m < ;) (vo)
ixof
)x(h))x(h9(l 6 9
8/12/2019 Limit of Function Theory_e
5/13
FI^GZ
"filosg`ufirpgysojs.ol" 5
(voo)ix
of e))x(h9( )x(h
9
(j)ix
of h(x) 6 I < ; iln
ixof (x) 6 M(i holote quiltoty), tgel
ixof Ph(x)V(x) 6 IIM.
Exifpde # > 1 Eviduite 1 ;x of x9)x9(
l
Zcdutocl 1;x
of
x
9)x9( l 6
;xof
9)x9(
9)x9( l
6 l
Exifpde # 5 1 Eviduite 1;x
of
=/x
9e x:
Zcdutocl 1;x
of
=/x
9e x: 6
;xof = :
x:
9e x: 6 8.
Exifpde # 8 1 Eviduite 1;x
of
:x
xsolxtil
Zcdutocl 1;x
of
:x
xsolxtil 6
;xof
:x
)xjcs9(xtil 6
;xof
:
=
x
=
xsol=.xtil
6;x
of
=
9.
x
xtil.
=
=
x=
xsol
6=
9.
Exifpde # 2 1 Eviduite 1;x
of
x:sol
x=sol
Zcdutocl 1;x
of
x:sol
x=sol6
;xof
x:sol
x:.
x:
x=.
x=
x=sol6
x=
x=solof
;x= .
:
=.
x:sol
x:of
;x:
6 9 .:
=
x:
x:solof
;x: 6
:
= 9 6
:
=
Exifpde # 7 1 Eviduite 1 xof
x
x
=9
Zcdutocl 1 xof
x
x
=9
6
x.x
=of
xe
6 e=.
Exifpde # ? 1 Eviduite 1 (o):x
of
:x
ee :x
(oo);x
of
xjcs9
)9e(x x
Zcdutocl 1 (o) [ut y 6 x :. Zc, is x : y ;. ^gus
:xof
:x
ee :x
6 ;y
of
y
ee :y: 6 ;y
of
y
ee.e :y: 6 e: ;y
of
y
9ey 6 e: . 9 6 e:
8/12/2019 Limit of Function Theory_e
6/13
FI^GZ
"filosg`ufirpgysojs.ol" 8
(oo);x
of
xjcs9
)9e(x x
6
;xof
=
xsol=
)9e(x
=
x 6
=
9.
;xof
=
xsol
x.
x
9e
=
=x
6 =.
Zedh prijtoje prcmdefs
Eviduite tge hcddcwolb dofots 1 -
(7) ;xof
x
x=sol(?) ;x
of
7x
7
>
xjcs
=
xjcs
>
xjcs
=
xjcs9
====
(9;)>
x
of
x>
x=sol9
(99)
;xof
x
?5 xx
(9=) xof (9 + i=)x sol
x= )i9(
m
, wgere i ;
Ilswers 1 (7) = (?) :=
9
(9;) nces lct exost (99) l ?
5
(9=) m
Use ch sumstotut ocl ol scdvolb dof ot prcmdefsZcfetofes ol scdvolb dofot prcmdef we jclvert
ixof h(x) oltc
;gof h(i + g) cr
;gof h(i g) ijjcrnolb is
leen ch tge prcmdef. (gere g os ipprcijgolb tc zerc.)
Exifpde # 9; 1 Eviduite>
x
of
xsol=9
xtil9
Zcdutocl 1 [ut x 6>
+ g
x>
g ;
;gof
g>
sol=9
g>
til9
6;g
of
gjcsgsol9
gtil9
gtil99
6;g
of
=
gjcs
=
gsol=
=
gsol=
gtil9
gtil=
=
6;g
of
=
gjcs
=
gsol
=
gsol=
gtil=
tilg)9(
9
6 ;gof
=
gjcs
=
gsol
=
g=
gsol
g
tilg=
tilg)9(
9
6 9=
6 =.
D ofots usolb expilsocl(i) ;i.........,
!:
ilx
!=
ilx
!9
ilx9i
::==
x
8/12/2019 Limit of Function Theory_e
7/13
FI^GZ
"filosg`ufirpgysojs.ol" 2
(m) ......!:
x
!=
x
!9
x9e
:=x
(j) l (9+x) 6 9x9hcr.........,>
x
:
x
=
xx
>:=
(n) .....!2
x
!5
x
!:
xxxsol
25:
(e) .....!8
x
!>
x
!=
x9xjcs
8>=
(h) til x 6 ......95
x=
:
xx
5:
(b) til-9x 6 ....2
x
5
x
:
xx
25:
(g) sol-9x 6 .....x!2
5.:.9x!5
:.9x!:
9x 2
===
5
==
:
=
(o) sej-9x 6 ......!8
x89
!>
x5
!=
x9
8>=
(a) hcr |x| 4 9, l ]0 (9 + x)l 6 9 + lx +=.9
)9l(l x= +
:.=.9
)=l)(9l(l x: + ............
(`) x9
)x9( 6 e
.............x
=>
99
=
x9 =
Exifpde # 99 1 Eviduite;x
of
=
x
x
x9e
Zcdutocl 1 ;xof
=
x
x
x9e 6 ;x
of
=
=
x
x9......!=
xx9
6=
9
Exifpde # 9= 1 Eviduite ;xof
:x
xsolxtil
Zcdutocl 1 ;xof
:x
xsolxtil 6 ;x
of
:
::
x
......!:
xx.......
:
xx
6:
9+
8
96
=
9.
Exifpde # 9: 1 Eviduite9x
of
9x
=)x2 :9
Zcdutocl 1 [ut x 6 9 + g
;gof
g
=)g7( :9
8/12/2019 Limit of Function Theory_e
8/13
FI^GZ
"filosg`ufirpgysojs.ol" 7
6 ;gof
g
=7
g9.=
:
9
6 ;gof
g
9.......=.9
7
g9
:
9
:
9
7
g.
:
99=
=
6 ;gof =
=>
96
9=
9
Exifpde # 9> 1 Eviduite ;xof
xsolxtilx
=
xxsol)x9(l
=
Zcdutocl 1 ;xof
xsolxtilx
=
xxsol)x9(l
=
6 ;x
of
x
xsol.
x
xtil.x
=
x.....
!5
x
!:
xx.....
:
x
=
xx
:
=5::=
6:
9+
8
96
=
9
Exifpde # 95 1 Eviduite ;xof
xtil
)x9(e x9
Zcdutocl 1 ;xof
xtil
)x9(e x9
6 ;x
of
xtil
......=
x9ee
6 ;xof
=
e
xtil
x6
=
e
Exifpde # 98 1 Holn tge vidues ch i,m iln j sc tgitxsolx
jexjcsmieof
xx
;x
6 =
Zcdutocl 1xsolx
jexjcsmieof
xx
;x
6 = .....(9)
it x ; luferitcr fust me equid tc zerc i m + j 6 ; m 6 i + j .....(=)
Hrcf (9) & (=),xsolx
jexjcs)ji(ieof
xx
;x
6 =
;x
of
.....!5
x
!:
xxx
......!:
x
!=
x
!9
x9j......
!>
x
!=
x9)ji(......
!:
x
!=
x
!9
x9i
5:
:=>=:=
6 =
;x
of
......!5
x
!:
x
9
.....)ji(!:
x)ji(
x
)ji(
>= 6 =
Zolje ].G.Z os holote,
8/12/2019 Limit of Function Theory_e
9/13
FI^GZ
"filosg`ufirpgysojs.ol" ?
i j 6 ; i 6 j , tgel9
....;i=; 6 =
i 6 9 tgel j 6 9Hrcf (=), m 6 i + j 6 9 + 9 6 =
Dofot wgel x Ol tgese types ch prcmdefs we usuiddy jiljed cut tge breitest pcwer ch x jcffcl ol luferitcr iln
nelcfolitcr mctg. Idsc scfetofe wgel x , we use tc sumstotute y 6x
9iln ol tgos jise y ;+.
Exifpde # 92 1 Eviduite xof x sol
x
9
Zcdutocl 1 xof x sol
x
96 x
of
x9
x
9sol
6 9
Exifpde # 97 1 Eviduite xof
:x=
=x
Zcdutocl 1 xof
:x=
=x
xof
x
:=
x
=9
6
=
9.
Exifpde # 9? 1 Eviduite xof
=xx:
5x>x:=
=
Zcdutocl 1 xof
=xx:
5x>x:=
=
6 x
of
:
:=
x
=9
x
:x
5
x
>
x
9
6 ;
Exifpde # =; 1 Eviduite xof
=x
=x: =
Zcdutocl 1 xof
=x
=x: =
([ut x 6 t
9, x t ;+)
6 ;tof
t
)t=9(
t
9.t=:
=
=
6 ;tof
)t=9(
t=: =
|t|
t6
9
:
6 : .
8/12/2019 Limit of Function Theory_e
10/13
FI^GZ
"filosg`ufirpgysojs.ol" 9;
Zcfe ofpcrtilt lctes 1(o) x
ofx
lx6 ; (oo) x
ofxe
x6 ; (ooo) x
ofx
l
e
x6 ;
(ov) xof
x
lx l
6 ; (v) ;x
of x(lx) l 6 ;
Is x ,l x oljreises fujg sdcwer tgil ily (pcsotove) pcwer ch x wgere is e x oljreises fujghister tgil ily (pcsotove) pcwer ch x.
(vo) lof (9g)l 6 ; iln l
of (9 + g)l wgere g ; +.
Exifpde # =9 1 Eviduite xof
x
9;;;
e
x
Zcdutocl 1 xof
x
9;;;
e
x6 ;
Dofots ch hcrf 9
(I) Idd tgese hcrfs jil me jclverten oltc;
;hcrf ol tge hcddcwolb wiys
(i) Oh x 9, y , tgel z 6 (x)y os ch 9 hcrf l z 6 yl x
l z 6
y
9
lx
hcrf
;
;
Is y y
9 ; iln x 9 lx ;
(m) Oh x ;, y ;, tgel z 6 xy os ch (;;) hcrf l z 6 yl x
l z 6
lx
9
y
hcrf
;
;
(j) Oh x , y ;, tgel z 6 xy os ch (); hcrf
l z 6 yl x
l z 6
lx
9
y
hcrf
;
;
(M) (9) type ch prcmdefs jil me scdven my tge hcddcwolb fetgcn
(i);x
of x
9
)x9( 6 e
(m)ix
of Ph(x)Vb(x) 0 wgere h(x) 9 0 b(x) is x i
8/12/2019 Limit of Function Theory_e
11/13
FI^GZ
"filosg`ufirpgysojs.ol" 99
6ix
of
)x(b.}9)x(h{9)x(h
9
9)x(h9
6 ixof
)x(b)9)x(h(
9)x(h
9
V)9)x(h(9P
6 )x(bV9)x(hPof
ixe
Exifpde # == 1 Eviduite xof
=x>
=
=
:x=
9x=
Zcdutocl 1 Zolje ot os ol tge hcrf ch 9
xof
=x>
=
=
:x=
9x=
6 exp
)=x>(:x=
:x=9x=of =
=
==
x 6 e7
Exifpde # =: 1 Eviduite
>x
of
(til x)til =x
Zcdutocl 1 Zolje ot os ol tge hcrf ch 9 sc>
x
of
(til x)til =x 6
x=til)9x(tilof
>x
e
6xtil9
xtil=)9x(tilof
=
>x
e
6 )>/til9(9
>/til=
e
6 e9 6e
9
Exifpde # => 1 Eviduite ixof
i=
xtil
x
i
=
.
Zcdutocl 1ix
of
i=
xtil
x
i=
put x 6 i + g
;g
of
i=
g
=til
)gi(
g9
;gof
i=
gjct
gi
g9 6
9
gi
g9.
i=
gjctof
;ge
gi
i=
.
i=
g
til
i=
g
of;g
e
6 =
e
Exifpde # =5 1 Eviduite ;xof xx
Zcdutocl 1 Det y 6 ;xof xx
l y 6 ;xof xl x 6 ;x
of
x
9x
9l
6 ;, isx
9
y 6 9
8/12/2019 Limit of Function Theory_e
12/13
FI^GZ
"filosg`ufirpgysojs.ol" 9=
Zilnwotjg tgecref cr squeeze pdiy tgecref1Zuppcse tgit h(x) b(x) g(x) hcr idd x ol scfe cpel oltervid jcltiololb i, exjept pcssomdy it x 6 iotsedh. Zuppcse idsc tgit
ixof h(x) 6 6 ix
of g(x),
^gel ixof b(x) 6 .
g
b
h
Exifpde # =8 1 Eviduite lof
=
l
VlxP....Vx:PVx=PVxP , wgere P.V nelctes breitest olteber huljtocl.
Zcdutocl 1 Re `lcw tgit, x 9 4 PxV x=x 9 4 P=xV =x:x 9 4 P:xV :x. . .
. . .
lx 9 4 PlxV lx (x + =x + :x + .... + lx) l 4 PxV + P=xV + ..... +PlxV (x + =x + .... + lx)
=
)9l(xl l 4
l
9r
VxrP =
)9l(l.x
lof
=
x
l
99
l
94 l
of=l
VlxP....Vx=PVxP l
of=
x
l
99
=
x4 l
of=l
VlxP....Vx=PVxP
=
x
lof
=l
VlxP....Vx=PVxP 6
=
x
I d o t er
Re `lcw tgit PxV 6 x {x}
VxrP
l
9r 6 PxV + P=xV + .... + PlxV6 (x + =x + :x + ... + lx) ({x} + {=x} + .... + {lx})
6=
)9l(xl ({x } + {=x} + .. + {lx})
=l
9
l
9r
VxrP 6=
x
l
99 =l
}lx{....}x={}x{
Zolje, ; {rx} 4 9, ;
l
9r
}xr{ 4 l
8/12/2019 Limit of Function Theory_e
13/13
FI^GZ
"filosg`ufirpgysojs.ol" 9:
lof
=
l
9r
l
}rx{ 6 ;
l
of=
l
9r
l
VrxP
6l
of
=
x
l
99
l
of=
l
9r
l
}rx{
l
of
=
l
9r
l
VrxP 6
=
x
