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Passing the limit through a continuous function
Math 8 2009, Chris Lyons
Here’s a helpful trick that allows you to “pass” a limit from outside a continuous function to the inside.
Proposition 1.
Suppose that
(i)
g
(
y
)
is continuous at
y
=
L
,
(ii)
lim
x
→
a
f
(
x
) =
L
.
Then
lim
x
→
a
g
(
f
(
x
)
)
=
g
±
lim
x
→
a
f
(
x
)
²
.
Before giving a proof, here’s the intuitive idea behind the statement: let
y
=
f
(
x
). Then as
x
→
a
we have
y
=
f
(
x
)
→
L
, and as
y
→
L
we have
g
(
y
)
→
g
(
L
). In other words: as
x
→
a
we have
g
(
f
(
x
)
)
→
g
(
lim
x
→
a
f
(
x
)
)
.
Now here’s a real proof, using the theorem that the composition of two continuous functions is again
continuous.
Proof.
Let’s make a new function:
F
(
x
) =
³
f
(
x
)
if
x
6
=
a
L
if
x
=
a
Then we know that
F
(
x
) is continuous at
x
=
a
because, by assumption (ii), we have
lim
x
→
a
F
(
x
) = lim
x
→
a
f
(
x
) =
L
=
F
(
a
)
.
So now by Theorem 3.5 in the book, since
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This note was uploaded on 07/19/2010 for the course MA 8 taught by Professor Vuletic,m during the Fall '08 term at Caltech.
 Fall '08
 Vuletic,M
 Calculus, Continuity

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