1
Comparing Growth of ln(x) and x 3
We have one more item on our original list of limits to cover; again we’ll look
at a slight variation on the original problem. We’re going to find:
lim
x→∞
This limit is of the form
ln x
x→∞ x1/3
lim
∞
∞,
ln x
.
x1/3
so we apply l’Hôpital’s rule to find:
=
=
=
lim
1/x
x→∞ 1 x−2/3
3
lim 3x−1/3
x→∞
(l’Hop)
0
We conclude that ln x grows more slowly as x approaches infinity than x1/3
or any positive power of x. In other words, ln x increases very slowly.
Question: When we discussed extensions of l’Hôpital’s rule, we learned
that we’re allowed to change some hypotheses. How many hypotheses can we
change at once?
f (a)
Answer: We can make any or all of the three changes listed. However,
g(a)
∞
0
,
−
,
or
.
must always be of the form ∞
∞
∞
0