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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
plugin bad why? Plugin
¥¥htiyt¥¥¥tyg¥-7
g 4 -2×
plugin bad.
+
picture =lim 1- =
-1
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