l36
Dijj
et
eutiutiu u
n
Secttn,oll 32.
ilimiits
For a more advanced example consider
Contrinulty
alad
M
i
once again di-rect subsiitution rvill not do,
Evidently
symbolized
j.
Direct substirution yierds
closer io
as
3,/(x)
becomes closer
= 13 , or/(x) -+ 13 as -r +
32'1. we
shalr say
jrg:/(-y)
=
r
tf
f(x)
can be made arbitrarily crose to
ll::::l
--^-
]tg53x
then.fair{y.easy
see that
-to
limit is 10,
value of the
if-r
approaches
(")
fim
Yrt'
(3-r +
L
z
Direct subsiiruiion yierds
,i
(which is yery near to zero), then
*
(whatever
* means).
J{r..ft"l
rla
s(x)] =
/(x) S(x)
,
lim
=x_)a
'-"
!g"f@)/g(x) =
=
-
106, rvhicl;
is iiot even
I
l#ge number of Limits vrithout r rucrr
flraboLu.
=Land _t r g(x) = IttI. Tlrcn
t_o
x+a"
r-
,{;/trl r 1im s(x) = L + kt.
,f(;r) .tim s&) = LM
x+o"..
4-a
;tg.f$)/fi"!{x)
=
(-
.
t,*
"mF,
--,-sqat
,provit)ed tc +0
Et,.
The proofof this theolem is son-rewhat technical aud we
shall not consi,ter.it hele.
Theonerm 32.3. (a)
then 3x + 4 approaches 10. Thus the
Example
lin x=a
x4o
(b1
lf f(.t)=c
*r"r,J'urr"*, ;;r';J;
;l
i'J,nriclr
then tim /(x)=c.
, ,
x)a"
L
Evaluate lirn (x2 + 4.r + 3).
x -+2
lVe have
jE
'l}"'
ff
t
*
However, using rhis rheorem ana ure two-rairiy
curailed.
"-u.ri"",
4) = 10.
fi- ;' -.t
,Jt.r_l
the nreaningles, ro*
-lim
x)o
':
[[(x)
F-
ri
C.
one mighr be.remprerr ro thi,,,r rhar
(dl lim /r/(x) = k lim /(-r) = ld- for ail k e
X-+a
x )a'
Sometimes. however, matters ate not so obvious. For
example consider
observ Lig
,
.
ofcourse, one's reac{.ion to this may.be that it is al] utteriy
obvious jndeed
in u'I@rr
manv cases
--- 'limits can be obrained bv direct ruuirii"tio* rrri;."""";;u;;;#, :
Iim (x2 - 4) =21 and lim^ (x + 3) = 1.
-
r+5'
,8.
(b)
0.
z,
10-6
lfheorery! 32.2, Suppose
thatx should be arrowed to ser croseland
cioser to aTt"^y::Lyhat,foilows
it any fashion, but that it shourd
neven actuarny nEacru o.-+
-
(32.1)
3.
io-a, uut without actuarly alrowing -r to be equar to a.
"nough
Consider for example
x.=
The following theorem ailows us ro compure a
to 13. This wi_ri be
(c)
whe'ever rve choose x crore
f
However,
lt IImIts
Deillmteion
ror- j .
rrre undefined
"91';=
positive, let al6ne near
above argumenr oniy makes (32.1) pJa-usibre - it does not prove
it. rn order ro give a
formal proof, we should need a.fblmar 'a.ri"ition. ihi;-;;;rd";;
;riiifi'airi"-ii i, ,rri,
slage, so we shall malce do wirh $re following
.i,jiirftf"*
"Ai.j
IT 9--- !-.
(xz +
4r + 3) =iim^ xr
=ltun^.r
*
,b+
$r
=4+B-r'3
. This problem ca,n be overcome by
=
='tg1G+1)=2.
I
h"'
ff
F-;'
X
..t
T'e
,l
,*{}{tl'i:i:f*
1
x+Q
r
,$,lttl
;:
is far.
_.
lrm *
Consider then
= il
f(4\
=15
"fA)
f\?.s-) : 12
rb'.st =14
12.3
f.\?.\ =
f\.l) = 13.2
f(z.ee) = t2.e8 7it.oi> = 13.02
lt would appear that as -r becomes
ri
bult
fi.om obvious hor,,, to rroceed.
Indeeditisnor.evencreartharr'eIimit.niir.utoii-dui*.i"i,',i,iprl'ir"i"i"r]
.,ii"',i.i,*,
It is obvious that/(3) = 13. However, the quesrion'we are interested in is not
what is the
value of /.v'rhel r:3, but what is the vaiqe orlrr,.n.r is ciose to 3.
consider a few
i'e'
sin ;r
does not exist.
of -r as x approaches a deFrnile ualue.
Let us consider a ve-ry simple exar4ple. Suppose/(x) Lr +
=
It is
,.
igb ,
As we saw in the last section,.it-is.frequendy necessary to consider the iaiue of
some
'
function
values.
(*
imits &nd rContinuitl,
|
15.
4"
3
,
(Theorern 32.2(a))
"J!1
* 4lim
.r + 3 (by 'Iher:r.ems 32.2(b)(ti1, 32.3(b))
iby'Xheorern 3e.S(t,j)
t,
Differentiation
i,,,:.auz,t|"
:i, ,l'
1r .l
.l
Liwits and Continwity
-r2 +
*
6uutuui"
3i.+ i
s
3tL.2,and3?,3 repeatedly; rve
obtain
Eremple
x2+3x+1
,,
lllTl-
,$"
:
Lranplet such
liry
' -a
as *iese are covered
,b11r
x
ilm
ta
DU
+ lir
,+
bg+
P(l:,:Sp:.nt* + .:. + anxn an:d
+O , then ull-='
ttm {Q = P-ftl
, +a Q.(x) O(a\
bP
Example
+ bza2
S.
e@) = b6 + b1x + ., +
:rmilar fashion to example
3.
vhat fris
f
r.
tlteorem savs is
."i,r.i ,rjir"ii!'J'""Ji:fi
a
The
orems 32.2 and 32.3 n
[til,Ti: iit*s
lffitolll
-*
Example
f@=++
,l
la:
!c
euough.
rvi rny
i.orLrlts
"tim
lirnirl r,f this
:
r' A --. () , rhen
l<ind
z
+
rJm
x- ,ll+t
a
--7j
fef,inaition
large enough.
fairly easy to evaluate.
_. -.rst
Fi
we note the follo.aing simpie
=
0"
(.r _
needs
t).
anot'er definition.
32,6."lg1,f(x)=*ifw.1 canmalce/(r)aslargeaqwepleasebytakingr
:_
If ii > 0 then
Please
.lge;tT7 =.,{|G-7tr
lim
=x+€
we sho'Id like ro say trrat this rimit is-infiiite.
This
by choosing.r
r
+1
"ry_#l_
The following results are fairly obvious
a-r.e
=o
;;"-;;;;i;i,i"til,g
Dividingbyxwehave_tim
Dividing byx we have
show'l]:!ll)-, 1.5, f (?) -2.4, (r,o)
f
=z.g.t[3and /(r00) =2.9997.
$ :::"#jil::{ ?:Hlf)o$qr"s .r*"..i,"J'croill,'o'i u, -r. becomes rarger and rarger.
would
YernltrQn
obscu'e,the
ittu.,
ro r.-i',ur["ti"'i'Jn'o*i'g
sf,"ewi;;;;ii;;;;:'
be made arbitrarily crose to
'ir
7.
Evatuare
Lj recr colnPutaLions
/(r) = L ii Ik)can
=-B-
Please note that rhe methoa of the last
three examples is only appricabre when
evaruating a
(or _r
--;. rt isoi
a iimtt as tends ro some finire
"o
given by
ri;ion, 32'5'
(32.2).
,t.
')*x-1+3x
of a fr
function it is frequently-usetut
vely largeJ}tr'ffi%T
jl. xample the function ro know what happens ro
values ofx. Consia",
i^..1
a#i,..,:??#
::N
=hh+=*.tu
J
r+6 1
lim'r
rorm can be evaluared bv direct
subsritution
sketch_ing the
rhe €raph
gaph
_lre,r
(ll'i,."::::!.n'ns
rot'
5
a
xla
)
t.
Dividing throughout by x we obtain
a
-l
1r
ruu (32.2)) =
.
6,
Evaluate lim
=" '
x-)-xz-X+3
he caoe. p(n)
=.1 is of particutar inteiesr In that case Theorem 32.4
says *rar if p is any
p
p. ,r],'lortuat
rl],'nornial dren
ilren lim p(l) = p(d.
t;,|
I
Dividing throughout by 12, we have
x+-" Jt--:, -l
xx4
of
rirn
to an artifice. Divid.ing the
-2. 2x + 7
!'-x]*3x2+x-5'
b^tm.If
Exanple
'
50 w€
"', ' resorr
Eva-luate 1;rn
,r;:
-= P(a)
alter repeared applications
d.O ,
.
t--
by the following theorem.
lilaeoremr 32.4'.. Let
.J@)
b2xz =
I
t
., x+ .I
'-:-L/
Irm;
,
=lim--=l=
_
I
x+6n
r+_ r I
+ btx +'tb2xz), where,Ds, 01,
b2 are constanrs.
* b2x\ =
fao + b.tx
-
and denominalor by x, we
.c,xa:nple 3.
tfn
. _r_ig1
+.d
V/e have, ,.
-
++
clearly we cannot merery substirute.r e,
-have
=
numerator
J
Eya.luale
[m
;-ge;
19
L or @ is replaced by
4.
Eva.luate
x+l
r+3
the results of Theorem 32.2 hold
wher.e a (bur nor
f"}t:i.tt
Usin6 Tlreorem
,:
t39
2.
r2..\
pi ,, = r+e
-F,t) = *.
tim (-r,,
.
]* i;. *), = _t+a
t+e
qote that jl-",
:tlP:t "*" is only used when raUcing abour
rcau
\J'.LJ
ilumner
a limit.
Z
.i
l$nrr*.
)
I)iJfere nliatio n
t90
![!xampl]
n
8.
. r5 -2.r2 r 5.r t
Evaluatc
- -2r't'l
"'*"**.'i-lirn '- Jtz
-This is the sarne as
,,."' _
lr!
r-
Contifi miu,orus lFrnnatuont
I
Defimitiom 32,3. A function/is comtinuous if, for iury
(.
F-1.'+t'-*4
re- ----n-+
3_Z+ E^
imits end Cantin*itjt
.
-
xxL
Exarnple
ln general we can maire ttre following statement:-
9.
Consider
P(*) = ao + atx + ... + anxn
QQ) = bs +bF
:nd
(a) if n>n thenlim
=-
hn f(x):J@)
Geometrically-this means that the graph of./has no'lurnps" or "holesl'in ir. Tl,e r:oflcepl.
'
may be most cJearly unde'stood byionsidering some lis..onilnuous f.nctions.
=llm-=@
Suppose
a,
+'.. + b,&n
lx+l when-r>b
when-r<0^
Ix
f@ :1
The graph
of/is
shorvn beior
whate an, b^ > 0'
ffi
(b)
P -"
=O
*ii0(x)
ifri<rr
(c) if n=m
then iim
**"*Lffi
=fi
X.
.
'l'hus if the highest power of x in the denorninator is lesftfan the highest power in the
numerator, thd ratio tends to infinity. If the reverse holds it tends to zero, while if the
highest power in each case is equai, then the iimit is the ratio of the coefficients of the
highest order terms.
One further kjnd of limit arises natuaIIy. A typical case is given by
..
Itm
I
,
we note that
32.7.
hnt
x+a "f(;r)
=
if we can make/(.r)
-
as large as we please
because the graph
a
It is knportant to note that we are choosing x to be near a, but
..
ltm
l1.r)=, 0.1, -0.0t ai.l
.{).0{)
ol/has a"jump" atx=0.
E,xatnple 10,
by taking ;r
Consider
close enough (but not equal to) a.
whetlrerx > a ot x < a. For example
.
)+0"
f(0.1) = 100, /(0.01) = I 04 ,/(-0. i) = i00 and/(- 0.01) = 1s+.
lVe should therefore like to say the the iimit is infinite. The following definition is then
natural combination of the ideas of Definitions 32.\ and32.6.
lDefirnirtiom
that lirn / (-r) = i
x+0-
different value frorn that obtained by approaching it irorn rhe righr. Tltur. lar fiorn
being equal to/(0), JrJrl" /(x) is nreartingless,-i.e. f is nor cdniinuor_r:. ,l-his is
1
*
One might be tempted to thinlc
Flowever, tatdngx=-0.1, -0.01, -0.001 we have
wrucn are not near lo l.
Thus lirn/(x.)' doesno[evenexist,ibrifyouapproach0lromtheleftyo,,t:hlaina
-,
x -+0 XL
Leuinc f(x) =
Ey definition/(0) = 1. Now what is iim /(jr)?
Talcingx=0.1, 0.01, 0.001 we have/(x)=1.1., 1.01, and 1.00i.
fx2 when.r+ l
/P; -{
t0 r,vhen -r = I
we are not specifying
r
1
r+0I does not exist. For when
x:0.01
we obtain 100, but whenx = - 0.01 we have
irlow rcca.ll (see remarir after Theorem 32.4) that
Iim P(x)
)t1562
100.
ts,
L'
Lt'
if P(-r) is a polynomial then
= P(c).
+ 1) = 4 - 4 +7 =
l(
i
For exantple
- 2-x
-
(4.t'o
)t'
t
Now/(i)
=S by definition. htowever,/1l.1 )
lim,
7.
This means that, in this case, one can merely substitute in when evaluating the Limit. This
pluDcrly is so pleasant that we give it a special name,
.LJl
/{x) -
=-.Zt,ltl.0t;
Il'r1 1'l - |
I +l
= 1.9261 u,,O
.
i
I
lirr.rit lin / (-r) rve clci .ot co'sider the r allie ol f'
)tl
-I
citself.) -Thus, since .l(1) * lim i(rr), I i:t n()t (,rcntinurru-r. This ir br:.rLrsre
@emember, when evaluating the
the graoh
of/has
a
I
"holc" irr il
I
{fiiH}mr
l(-
I
iri
rr*.{f:ilrff