l'H
os
pital's Rule: A Classic Example
(
29
1
Let's compute the limit:
lim 1
Notice, if I plug in the " ", we, formally, get:
1
1
lim 1
1
1
0
1
But recognize, the base isn't quite 1.
x
x
x
x
x
x
→ ∞
∞
∞
∞
→ ∞
+
∞
+
=
+
=
+
=
∞
And the exponent, of course isn't even a real
number, so what's going on?
Well, any number slightly larger than 1, to a large power, would expand to
.
Conversely, any number slightly smaller than 1 wou
∞
ld shrink away to zero.
And of course, an
value, taken to any power, even a very large one, is still 1.
So as it stands, it appears we have an
.
Now, is there any way to change th
exact
indeterminate form
is indeterminate form into one we know how to
deal with?
Well, since the problem is with the exponent, let's bring it down with a lnoperation.
1
If we let
lim 1
then
ln( )
ln lim
x
x
x
y
x
y
→ ∞
=
+
=
1
1
1
1
limln
1
lim ln 1
Again, plugging in our " ", then
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 Spring '11
 CHILDS
 Calculus

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