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