SECTION 5.1 Quick Survey of Limits 259
This result we summarize as
l’Hˆopital’s Rule. Let f and g be functions differentiable on some
interval containingx=a, and assume thatf′andg′are continuous at
x=a. Then
xlim→af(x)
g(x)=
xlim→af′(x)
xlim→ag′(x).
As a simple illustration, watch this:xlim→ 32 x^2 − 7 x+ 3
x− 3=
lim
x→ 3
4 x− 7xlim→ 31= 5,
which agrees with the answer obtained algebraically.
In a similar manner, one defines∞/∞indeterminate forms; these
are treated as above, namely by differentiating numerator and denom-
inator:
l’Hˆopital’s Rule(∞/∞). Letf andgbe functions differentiable on
some interval containingx=a, thatxlim→af(x) = ±∞= limx→ag(x), and
assume thatf′andg′are continuous atx=a. Then
xlim→af(x)
g(x)=
xlim→af′(x)
xlim→ag′(x).
There are other indeterminate forms as well: 0 ·∞, 1 ∞,and∞^0.
These can be treated as indicated in the examples below.
Example 1. Compute lim
x→ 0 +
x^2 lnx. Note that this is a 0 ·∞indeter-
minate form. It can easily be converted to an ∞∞ indeterminate form
and handled as above:
lim
x→ 0 +
x^2 lnx = lim
x→ 0 +lnx
(1/x^2 )l’H= lim
x→ 0 +1 /x
− 2 /x^3= lim
x→ 0 +−x^2
2