Calculus- Early Transcendentals, 2021a

192 Applications of Derivatives
Definition 5.33: Limits of the Indeterminate Forms 0 0 and ∞ ∞
f (x)
A limit of a quotient lim
x→a g(x) is said to be an indeterminate form of the type 0 0 if
both f (x) → 0 and g(x) → 0 as x → a.
Likewise, it is said to be an indeterminate form of the type ∞ ∞ if
both f (x) →±∞ and g(x) →±∞ as x → a.
Note that the two ± signs are independent of each other.
Theorem 5.34: L’Hôpital’s Rule
f (x)
For a limit lim
x→a g(x) of the indeterminate form 0 0 or ∞ ∞ ,
f
if lim
′ (x)
x→a g ′ (x)
exists or equals ∞ or −∞.
f (x)
lim
x→a g(x) = lim f ′ (x)
x→a g ′ (x)
This theorem is somewhat difficult to prove, in part because it incorporates so many different possibilities,
so we will not prove it here.
We should also note that there may be instances where we would need to apply L’Hôpital’s Rule
f
multiple times, but we must confirm that lim ′ (x)
x→a g ′ (x)
is still indeterminate before we attempt to apply
L’Hôpital’s Rule again. Finally, we want to mention that L’Hôpital’s rule is also valid for one-sided limits
and limits at infinity.
Example 5.35: L’Hôpital’s Rule
x 2 − π 2
Compute lim
x→π sinx .
Solution. We use L’Hôpital’s Rule: Since the numerator and denominator both approach zero,
x 2 − π 2 2x
lim = lim
x→π sinx x→π cosx ,
provided the latter exists. But in fact this is an easy limit, since the denominator now approaches −1, so
x 2 − π 2
lim = 2π
x→π sinx −1 = −2π.
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