L'Hopital's Rule can be used to evaluate limits that result in indeterminate forms like 0/0 and ∞/∞. It works by taking the derivative of the top and bottom functions and re-evaluating the limit. Two examples are worked out in the document, applying L'Hopital's Rule multiple times until the limit can be determined. The rule should only be applied to quotients that are indeterminate forms, and some forms like 0*∞ are specifically identified as indeterminate and able to use the rule.

1. L’Hôpital’s Rule
If the circumstances are right, L’Hopital’s Rule can
speed up limits.

2. Recall that there are some functions that when finding limits produce the
indeterminate forms 0/0 or ∞/∞. When these come up, they do
not guarantee a limit exists, nor what that limit is.
Some examples: 2x2 − 2 3x 2 − 2
lim lim 2
x →−1 x + 1 x →∞ 2 x + 1
Rewriting with algebraic techniques can handle some of them, for example
dividing out techniques, rationalizing denominators or numerators, or
using the squeeze theorem.

3. This extended Mean Value Theorem and L’Hôpital’s Rule are proved
in Appendix A at the back of the book.
Consider the Mean Value Theorem applied to f(x) and g(x).
f (b) − f (a )
f '(c) b−a
=
g ' ( c) g (b) − g (a )
b−a

4. This rule is used incorrectly if you apply the quotient rule to f(x)/g(x).

5. Let’s look at using L’Hôpital’s Rule on the examples we looked at in
the beginning.
2x2 − 2 Direct substitution achieves the indeterminate form
lim 0/0. Letting f ( x ) = 2and − 2
x2
x →−1 x + 1
g( x) = x + 1
f ' ( x) 4x
= lim = lim = −4
x →−1 g ' ( x ) x →−1 1
3x 2 − 2 Direct substitution achieves the indeterminate form ∞/∞.
lim 2
x →∞ 2 x + 1
Let f(x) be the numerator, and g(x) be the
denominator.
3x 2 − 2 6x 6 3 (notice we just kept applying
lim 2 = lim = lim = L’Hopital’s Rule until conditions
x →∞ 2 x + 1 x →∞ 4 x x →∞ 4 2 didn’t apply or we were done)

6. 0 , ±∞ , ∞ − ∞, 0 ⋅ ∞, 00 , 1∞ , and ∞ 0
The forms 0 ±∞
all have been identified as indeterminate. All but the first two would have
to be rewritten in quotient form before L’Hôpital’s Rule applies
There are similar forms that have been identified as determinate.
L’Hôpital’s Rule doesn’t apply.
∞+∞→∞ Limit is positive infinity
−∞ − ∞ → −∞ Limit is negative infinity
∞
0 →0 Limit is zero
−∞
0 →∞ Limit is positive infinity

7. Caution! L’Hôpital’s Rule can be applied ONLY to quotients leading
to the indeterminate forms 0/0 or ∞/∞!
Find the following limit:
x2 − x − 6
lim
x →2 x−2

8. Find the following limit:
1 (4 + x) − 1 2
4+ x −2 0
lim = = lim 2
x →0 x 0 x →0 1
1 1
= lim =
x →0 2 4 + x 4

9. 8.7 p574/ 1-11 odd,19, 23, 31, 39, 51, 76, 77, 79