Math 151: 4.8 Linearisation to Understand l’Hopital’s Rule
Linearization as Motivation for L’Hopital’s Rule
s
Write the equations for the following
graphs and write the ratio
.
,
1
x − 2) +
(
3 . The limit is indeterminate because
Now evaluate limit lim 3
2sin ( x − 3)
1
1
f ( x ) = − ( x − 2) + ⇒ f (3) = 0 and g ( x ) = 2sin ( x − 3) ⇒ g (3) = 0 . Since x gets very
3
3
close to 3, one way to determine it (the limit) would be to approximate f ( x ) and g ( x )
with the linearizations at x = 3 , L ( x ) and L ( x ) .
−
1
2
x →3
2
f
g
The tangents to each function at x = 3 are
()
f x =−
1
2
x − 2) + ; L ( x ) = − x + 2
(
3
3
3
1
Note that −
2
f
2
3
x +2= −
2
3
()
(
) ()
g x = 2sin x − 3 ; L g x = 2x − 6
(x − 3) and 2x − 6 = 2(x − 3) .
© Raelene Dufresne 2012
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Math 151: 4.8 Linearisation to Understand l’Hopital’s Rule
Here are some different views of the functions and their linearizations:
Would you expect these approximations to be very good? Why or why not?
Thus, we can rewrite lim
−
x →3
earlier for the ratio
( ).
L (x )
(
3
1
)
2
x −2 +
(
)
2sin x − 3
1
3 = lim
x →3
2
x +2
2
(
)
x −3
1
3
3
= lim
= − as we saw
x →3 2 x − 3
2x − 6
3
−
−
(
)
Lf x
g
© Raelene Dufresne 2012
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Math 151: 4.8 Linearisation to Understand l’Hopital’s Rule
This example helps us to visually understand L’Hopital’s Rule, by which we can rewrite
lim
x →a
( ) = lim f ′ (x ) …
g (x )
g′ (x )
f x
x →a
… as long as
• f and g are differentiable on an open interval containing a,
•
•
Finding the derivatives:
By L’Hopital’s Rule,
lim
x →a
( ) = lim f ′ (x ) :
g (x )
g′ (x )
f x
x →a
The linearization method gave us the same result as L’Hoptial’s Rule!
We can conceptualize L’Hoptial’s Rule as lim
x →a
()
()
( )
( ) = lim f ′ (x ) = lim L (x ) = L ′ (x ) ,
g (x )
g′ (x )
L (x ) L ′ (x )
f x
x →a
x →a
f
f
g
g
where Lf x and L g x intersect at a, 0 .
While this example is not trying to suggest that
d
( (
)) (
2
d ⎛ 1
1⎞
2
−
x
−
2
+
⎜
⎟ = − x − 3 nor
dx ⎝ 3
3⎠
3
(
)
(
)
)
2sin x − 3 = 2 x − 3 , we do elicit the concepts of local linearity and linear
dx
approximation to reinforce differentiability (a requirement of L’Hopital’s Rule) and to
visualize what is happening at a local level (when x is “close to” 3) for L’Hopital’s Rule.
that
© Raelene Dufresne 2012
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