7/23/2019 Comment lever une indtermination dans une limite (1/2)
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1/1
AokdfIomotfs&Kcltoluot
MtdcjfsftNllnifs
dttp7//fxcs8mntd.arff.ar/
[nvcor ifvfr ulf oljtfrmolntocl jnls ulf iomotf.
Xnppfi jf kcurs
Prcprots
In iomotf julf aclktocl pciylmf fl fst enif in iomotf jf scl
tfrmf jf pius dnut jfer. In iomotf julf aclktocl rntocllfiif flfst
enif in iomotf ju quctoflt jfs tfrmfs jf pius dnut
jfer.Prcprots
[o 0, ;5 (jmcltr jnls if kcurs), iomx+
xfx = 0 iomx+
fx
x= + iom
x0+x il x= 0 iom
x+
il x
x= 0 iom
x0
fx >x
= >
Mtdcjf
Pcur tujofr in iomotf jul pciylmf fl , oi anut mfttrf if tfrmf
jf pius dnut jfer fl anktfur, cu hofl utoiosfrin prcprot >. Pcur
tujofr in iomotf julf aclktocl rntocllfiif fl, oi anut anktcrosfr
if lumrntfur ft if jlcmolntfur pnrifur tfrmf jf pius dnut jfer, cu
hofl utoiosfr in prcprot 8. [o in aclktocl kcltoflt ulf acrmf
fxpclfltofiif cu icenrotdmoquf, oi anut anktcrosfr pnr if tfrmf
jcmolnlt flutoiosnlt ifs krcossnlkfs kcmpnrfs (prcprot ;). Kfs
reifs sclt uloqufmflt vninhifs jnls if kns jfs iomotfs ollofs
!Icrsquoi anut tujofr ulf iomotf flul pcolt n jnls if kns julf
aclktocl rntocllfiif, oi anut elrnifmflt anktcrosfr if jlcmolntfur
pnr (x n) knr nfst scuvflt in vnifur quo nlluif if jlcmolntfur.
(Qcor if knikui jf in ;mf iomotf jf ifxfrkokf >).
Xfmnrquf 7 Pcur ifs pcolts > ft 8 jf in mtdcjf, oi anut
provoieofr in anktcrosntocl iutoiosntocl jfs prcprots(knr vcus lf
ifs jmcltrfz pns) nl jnvcor tcus ifs pcolts in qufstocl.
Fxfrkokf >
Knikuifr ifs iomotfs suovnltfs 7 iomx+
x8 + 8 x8
iom
x
x8 >8x8 + 6
iomx>
x8 >x;
;x+ 8
.
Fxfrkokf 8 (Lcuvfiif Knijclof 800:)
Cl kclsojrf in aclktocl ajlof sur ioltfrvniif Xpnr 7 a(x) =
x
fx x . Jtfrmolfr in iomotf jf a fl +.
Fxfrkokf >
Cl vcot hofl quf soi cl knikuif jorfktfmflt in iomotf fl+, if
rsuitnt fst oljtfrmol 7+ = 3. Pcur ifvfrioljtfrmolntocl, cl
nppioquf in mtdcjf fl anktcrosnltpnr if tfrmf jf pius dnut jfer
7
iomx+x8 + 8 x
8
= iomx+x8
>
x> + 8
x8 >.
Cl pfut jscrmnos knikuifr in iomotf jf kdnquf mfmhrf 7
iomx+
>
x
> +
8
x8 >
=
0
> + 0 >
= >.
Jclk iomx+
x8 + 8 x8
= iom
x+x8 = .
x8 fst if tfrmf jf pius dnut jfer jf in arnktocl. Clanktcrosf
nicrs if jlcmolntfur ft if lumrntfur pnr x8 7
iomx
x8 >8x8 + 6
= iomx
x8
> >x8
x8
8 + 6x8
cr iomx
>x8
= 0.
Jclk iomx
x8 >
8x8
+ 6
= >
8
.
> fst ulf rnkolf ju lumrntfur ft ju jlcmolntfur,kfst pcurquco
oi y n oljtfrmolntocl 7
0
0 = 3. Oi anut jclk
anktcrosfr fl dnut ft fl hns pnr (x >) 7iomx>
x8 >x; ;x+ 8 = iomx>
(x >)(x+ >)(x >)(x 8) = 8.
Fxfrkokf 8
Cl vcot hofl qufl + in iomotf ju jlcmolntfur vnut+ = 3. Oi y n
jfux mnlorfs jf anorf oko 7 Cl kdcosot jf anktcrosfr pnr if tfrmf
jcmolnlt fx fldnut ft fl hns jf in arnktocl 7
x
fx
x
= fx
fxxfx
(>
xfx).
Flsuotf cl utoiosf ifs Krcossnlkfs Kcmpnrfs 7
iomx+
xfx = 0. Cl chtoflt jclk iomx+
x
fx x = 0.
Cl pfut nusso kdcosor jf anktcrosfr pnr x 7x
fx x = x
x
>fx
x >
. Cl utoiosf tcugcurs ifs Krcos-
snlkfs Kcmpnrfs scot 7 iomx+
fx
x = +.Cl rftrcuvf hofl 7 iom
x+
x
fx x = 0.
