Indeterminate forms and L' Hospital 's rule
Idea : suppose we want to analyze a limit of the
form find gE¥T where Him fix) = me gcx)= as
.
This does not necessarily mean that Iim §y = I.
F is an indeterminate form and many things could happen .
How to tell ? We will see one way to do this today
as well as for other indeterminate forms like O_O .
Examples of 8- indeterminate forms :
sink)Iim I
= lX-70
x'-xlim Ii
-
- t.is#I=fi;nxIi-- EX -71
Iim = ?x - l
X -71
Examples of E indeterminate forms :
t.im.tt#-t:m.IIiiiiEi=:!?:fi.=o=oex
Iim Tooox-70 X
lncx)Iim Fix→cs
L' Hospital 's Rule : suppose that f and g are differentiable and
g'Cx ) -10 near a (except possibly at a ). Suppose also that
finna LCH - Ijm gcx ) -- O
OR
finna LCH -
- finna gcxl -- too .
Then
t.int#--t:magEE.T .
Note : still true if we replace finna by Liam , tim. .
.
lim limx→ a-
'
x→ at"
fa
l"y- yea,=µa,c×.a,
y-gcat-gtakx.at
gcx)
fa but fca) -- gca)-
- O
so y= L'Ca)Cx - a )
y=g'(a)Cx- a)
gcx)' Limal'cakx - a )
and finnag'Caux- a)
= Eng'Ca)
Example : find Lim,¥4 .
Ans :
Example : find xhjm,4¥ .
Ans : Iim lncxl = Incl ) = O,
lim x - I = I - 1=0x-71 X -71
SO I indeterminate form
t.i.m.tn#=tisi:I;EIIst=tnimt--ti.atx-- I
e.×
Example : find Lim T . Lim ¥ , fishAns :
exExample : find Lim T . Lim ¥ , fishAns : Ijm execs
, Lim x = find x' =L;z x""
-
- as
% indeterminate form
tins # -
- tins E -
- t.im.= -
Is indeterminate form
← again
tim.
' Ii:s÷÷E -
- tins -
- ti: :¥Esx
= lim e- = co
x-702
Example : find limInde
x→ asExt
Ans :
In Cx)
Example : find lim -
x→csTxt
Ans : Ising lncxl -- Ling'It = -
Fs indeterminate form
t.im.
. Him. iii. t.im. ¥i- ti:
"
= lion106×+114"
"o y%-
- times 10C""
I.im/oCx-"'+ x
"" )""
(Cy)-- 10,1%
= locating x- ' "
+ x''"9)""
= loco -101%0=0 is
continuous
Example : find II.mo
Ans :
1-an Cx) - XExample : find Limo 7,3Ans : Iim tan Cx) - x = lim x' = 0 Of indeterminate form
X-70 x -70)
d
tancxl - X DIE tank) - X) see 'Cx ) - I
Limo 2×3 = lim = limx-70 x -70 3×2
d¥[x3]
f- indeterminate form again
seicx ) - I ddxtseicx) - I ] Zseccx ) Eseccxi]Iim = Iim = limx -70 3×2 x -70 day [3×2] to Gx
Zsecicxltancx)= lim O_O indeterminate form again !x-70 f×
'
Example : find Limo j×Ans : t.im
-
- fi;mosee;
-
t.fi#2seicxItancxl6x=1imdxdE2sec4xltanCx1]to ¥ [ Gx)
= Yim2 ( Zseccxlseccxltancxltanlx) + see
" Cx ))x-70 6
2sec4x) tan-
Cx) t see"Cx ) Otl
= lim=
3-= ¥
x -70 3
Another way on next slide
1-an Cx) - xExample : find Limo 7,3Ans :
mo =L ;mosee;
- t.fi#2seicxItancx)(x
= Efim ÷×, sing' = 's ¥%÷*, t.im.
= 's f- I = I
Example : find limsink)
x→ it-
l - cos Cx )
Ans :
Example : find limsink)
x→ #-
I - Cosa )
Ans : Iim sink) = sink) - Ox-7T
-
Iim I - coscx) = I - cost) =Lx-7T
-
NOT AN INDETERMINATE FORM
sink)Iimx→*
-
I - cosa )= € = O
So far we have only discussed indeterminate
quotients ( Eor F )
.
We also have indeterminate products such as
Iim xlnlxl.
Note lim x -
- OX-70 X-70
Iim lnlxl = - co
x-70
Indeterminate form O - o.
Who wins ?
Idea : write fcxlgcx) -- = 7can
This turns a O - co indeterminate form into a
€ or Is indeterminate form so we can
use L'Hoapital 's rule.
Note : how to choose between and ?
Choose the one with the easier derivatives .
Example : find Yim. xlnlxl
Ans :
Example : find him xlnlxl.
X-20-
Ans : lnlxlxlnlxl = Is= -
1- IInlxl X
Teasier derivatives
Iim lnlxl ' - cox- o
-
⇒ I indeterminate formI
-as
Iim I = - co
x-soxtiom.xmxkt.io. =L;'s.FI#iij--*iom..IE=t:mo.-x--o
What else is there ? Indeterminate quotients ,
indeterminate products , and indeterminate differences.
Lima La )-
- finna gcxl = -
What is #ma LCH -g Cx ) ? (co- co indeterminate form)
Example : Iim x'- x = co
x-70
xhjms Txt - Fi = 0
To apply littopital 's rule , turn it into a quotient .
Example : find lim sects) - tank )x -7¥
Ans :
Example : find lim seicx) - tank )xx I2
Ans : Iim seicx) -- lim ÷×, = co
* I +→ I2 2
tank --
×!i÷ = -
ofco - as indeterminate form
←
×!inE seek)- tank) =¥n¥H
= IimCOSYX ) ⇒ Iz
Zcoscxlc- sin (x ) )
= Iim I = IxaE
Second ans : sin'Cxltcos4x) - I
But wait,
there's more ! Indeterminate powers!
x!'T exist
"
t.
×! 'T fix) -
- O and
×! inangcxl =o (type 00 )
2.
×!i7 fix) = O and Hiran gcxl =o (type 5)
3.
×! 'T fix) = I and
×! inangcxl = -
( type 10 )
But wait,
there's more ! Indeterminate powers!
tina LHP"
= Iim egad head*a
1-×! 'T fix) -
- O and
×! inangcxl =o (type 00 )
2.
×!i7 fix) = O and Hiran gcxl =o (type 5)
3.
×! 'T fix) = I and
×! inangcxl = -
( type 10 )
I
Example : find Limo, C ' - x )"
Ans :
I×
Example : find lim Ci - x )x→ Ot
Ans'
.
+Ling Ci -x ) - I
, Iim,
I - o, type to
limo, c ' - x = ! .im,
e'¥"
I -X = elna-X)
limo, k¥1 is E
= Iim Extinct -x))wot Eis
' x'thot'¥ -
- finna = - i
so t.im,
C ' - x =
mo,e'¥
"
= eTo
= e-'=
.
368 # I
Example : find lim xxx→ Ot
Ans :
Example : find lim xxx-70-1
Ans'
. lim x -
- O,
0° typeX-Sot
Lion xx -- Isna e"""
..
Ink)To type
X-
- e←
I
¥ Elnlxl] I
finna xlnlx) = finna'
II = lim - = Ising,I× Hot da [¥ ] Xz
=
mot- x = 0 so Lino, x'
'
= tyg, e"""
= et 's'F×h↳)
= e° = I
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