4-4: L’HOSPITAL’S RULE

1. Compare the following two limits:

(a) limx!4

2x2 � 5x� 12

x2 � 3x� 4=

(b) limx!4

ln(x� 3)

4x� x2=

2. L’Hospital’s Rule

1 Section 4-4

t.ie:1#=timEIlYII;Ima2x'÷=¥X -74 f-7 y

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¥¥htiyt¥¥¥tyg¥-7

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picture =lim 1- =

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ya-

:

×→4 4 - 8 4

t.IM#II:Isi::like tangent. lines close✓

✓y=4x-x2 to # 4

•

. If xhjmafixtoandxhjmagtnoithenxljma ftp.xhjmaf#y

provided the limit on the right exists .

( oristx )

•fcx ) Ii

. If limflxt '±o and xtimaglxktns , then lim gTp=×jmafgq¥}x→a

x→a

provided the limit on the right exists .

Loris Is )

3. (Some routine examples.) Evaluate the limits.

(a) limx!(⇡/2)+

cosx

1� sinx

(b) limx!1

(lnx)2

ln(x2)

(c) limx!5+

ex � 1

x� 5

(d) limx!1

ex

x2

4. L’Hospital’s Rule can address other indeterminate forms.

2 Section 4-4

¥←Y¥+ t.su#sIE+anx=+a

aforms

y.ly#yetanxx=Tzex.ewm.YyIIistzI=xI*o

= +x

+form e¥ .

L' Hospital 's does not apply .

'

as ×→5t, x. s → ot

;as x→5+ , eh I → e5 - l

.

ElimI# gig

.¥ = tying exes

.

y×→ - ¥

form 7. formng

like o.rs,

x -x ,10

,0°

,xD

5. Examples to demonstrate.

(a) limx!0+

x lnx

(b) limx!1

⇣1 +

a

x

⌘bx

6. Examples for you.

(a) limx!1+

x1

1�x

(b) limx!1

x3/2 sin

✓1

x

◆

(c) limx!0+

✓1

x� 1

ex � 1

◆

3 Section 4-4

,III '¥=I;+Y¥±9.IE#mo+.x=o

.

Tform OEX)form¥

x

E form ,I @If y=(1+axt)b×,

then lny=bxln( ltax' ) .

Consider lime bxlncltax ')= line bln(Hax")_ →

×→x X→x x- '

=

€C form P

I;+C⇒nxstiy+I*±¥×In+¥=- i

2 formfo

Cha=

lim Sind # 1in ask.IE-3/2 - z -512

x→x × ×→x zxI form X° 0

I form Of

=lim Zzxtwsl 'k)=x . ×Xtx @ e - l

eh - ×- line -= lim - - 16×-1)+×e×

, ,x→ot ×( ex . i ) Hot

,

x - x T form #form Of

×

# lim = =L

x→o+ e×+te×+xe× 2

Consider lim b×1n(|+a×D= line blncltax" )

×→x X→x #c form %

HO b- .- aE2

= lim Hay = lim *#=ab×→x - IZ ×→a ltax

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