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2018 L’Hopital’s Rule. BC Calculus. FINDING LIMITS. Rem: Always start with Direct Substitution. Method 1: Algebraic - Factorization. Method 2: Algebraic - Rationalization. Today is the Day!!!. Method 3: Numeric – Chart (last resort!). Method 4: Calculus. - PowerPoint PPT Presentation
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2018 L’Hopital’s Rule
BC Calculus
Rem: Always start with Direct Substitution
Method 1: Algebraic - Factorization
4
2 04 0x
xLimx
Method 2: Algebraic - Rationalization
3
1
1 01 0x
xLimx
Method 3: Numeric – Chart (last resort!)
3
0
1 00
x
x
eLimx
Method 4: CalculusTo be Learned Later !
FINDING LIMITS
Today
is
the
Day!!!
L’Hopital’s Rule
If produces the Indeterminant Form, ( or ),
then
Provided the limit exists and . (Also )
( )lim( )x a
f xg x
( )lim( )x
f xg x
( ) ( )lim lim( ) ( )x a x a
f x f xg x g x
00
( ) 0g x
CAUTION: Think of this as the comparison of two functions
NOT a Quotient Rule << NOT Bo-d(To) >>
IE> and
2x 2x
EXTRA: Proof L’Hopital’s Rule
( ) ( )'( ) limx a
f x f af xx a
Working backwards:( ) ( ) ( ) ( )lim'( )lim = lim( ) ( ) ( ) ( )'( ) lim
( ) ( ) = lim ( ) ( )
x a
x a x a
x a
x a
f x f a f x f af x x a x a
g x g a g x g ag xx a x a
f x f ag x g a
( ) 0= lim( ) 0
( ) = lim( )
x a
x a
f xg x
f xg x
( ) ( )'( ) limx a
g x g ag xx a
lim ( )( ) 0L'Hopital's Rule requires: lim therefore ( ) & ( ) 0
( ) lim ( ) 0x a
x ax a
f xf x f a g ag x g x
Does it Work?
2
6
3 186lim
x
x xx
2
2
3 2 145 6lim
x
x xx x
lim𝑥→ 6
𝑥2−3 𝑥−18𝑥−6
=00
lim𝑥→6
2𝑥−31 =9
lim𝑥→6
(𝑥−6)(𝑥+3)(𝑥−6)
=9
+∞+∞
lim𝑥→∞
3 𝑥2
5 𝑥2Leading term test
= ∞∞
= 35
befo
re
befo
re
By L’H
opita
l
By L’H
opita
l
L’Hopital’s Babies
EX:0
sin(4 )limsin( )x
xx
EX:ln( )lim
x
xx
¿00
lim𝑥→ 0
cos (4 𝑥 )(4)cos (𝑥)
=¿¿41
¿∞∞
lim𝑥→∞
1𝑥1
lim𝑥→→∞
1
𝑥 = 1∞
=0
Which grows faster?Bottom goes to 0Top goes to ∞
By L’H
opita
l
By L’H
opita
l
Repeated use of L’Hopital’s:
EX:20
1limx
x
e xx
¿
00
lim𝑥→0
𝑒𝑥 (1 )−1𝑥❑ =¿ ¿
00
lim𝑥→ 0
𝑒𝑥
2=¿¿
12
By L’H
opita
l
Repeated use of L’Hopital’s:
lim𝑥→∞
2𝑥
𝑥2∞∞
lim𝑥→∞
2𝑥 ¿¿¿
lim𝑥→∞
2𝑥 ¿¿¿∞2 =¿∞
By L’H
opita
l
By L’H
opita
l
L’Hopital’s Rule
Other Indeterminant Forms: << Rem: LIMITS not values! >>
0 00 , 0 , 1 , ,
Known Limits: 0, 0, 0,0 0c c
Convert to L’Hopital’s forms
Converting to L’Hopital’s forms:
EX: 0
lim ln( )x
x x
Converting to L’Hopital’s forms:
EX: 1
0lim sin ( ) csc( )x
x x
Converting to L’Hopital’s forms:
EX:0
1 1limsin( )x x x
Converting to L’Hopital’s forms:
EX:1lim 1
x
x x
Summary:L’Hopital ‘s Forms:
Converting:
Multiplication:
Subtraction:
Exponential:
Other Properties:
Last Update:
• 11/22/11
• Assignment: p. 450 # 1- 23 odd
Turkey Quiz !
• Find the Limit:
1)
2)
3)
1
ln( )1x
xLimx
2
x
x
eLimx
0sin( ) ln( )
xLim x x
