§7.8–Indeterminate forms and L’Hôpital’s rule
Mark Woodard
Furman U
Fall 2010
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
1 / 11
Outline
1
The forms “0/0” and “∞/∞”
2
Examples
3
Indeterminate products: “0 · ∞”
4
Indeterminate differences: “∞ − ∞”
5
Indeterminate powers: “00 ”, “∞0 ” and “1∞ ”
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
2 / 11
The forms “0/0” and “∞/∞”
Definition
We say that the limit
lim
x→a
f (x)
g (x)
is of the form “0/0” or “∞/∞” if
lim f (x) = lim g (x) = 0
x→a
x→a
or
lim f (x) = lim g (x) = ±∞,
x→a
x→a
respectively. These are called indeterminate. In this context, a can be any
of a+ , a− , ±∞.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
3 / 11
The forms “0/0” and “∞/∞”
Theorem (L’Hôpital’s Rule)
If limx→a f (x)/g (x) is an ideterminate form of type “0/0” or “∞/∞” and
if
f 0 (x)
= L,
lim 0
x→a g (x)
then
lim
x→a
Mark Woodard (Furman U)
f (x)
= L.
g (x)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
4 / 11
The forms “0/0” and “∞/∞”
Theorem (L’Hôpital’s Rule)
If limx→a f (x)/g (x) is an ideterminate form of type “0/0” or “∞/∞” and
if
f 0 (x)
= L,
lim 0
x→a g (x)
then
lim
x→a
f (x)
= L.
g (x)
Remark
When facing an indeterminate form, students will often write:
f (x)
f 0 (x)
= lim 0
x→a g (x)
x→a g (x)
lim
which is, strictly speaking, wrong. L’Hôpital’s rule states that these are
equal only when the limit on the right exists.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
4 / 11
Examples
Problem
Evaluate the following limits:
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
5 / 11
Examples
Problem
Evaluate the following limits:
sin(x)
limx→0
x
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
5 / 11
Examples
Problem
Evaluate the following limits:
sin(x)
limx→0
x
x2 − 1
limx→1
x −1
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
5 / 11
Examples
Problem
Evaluate the following limits:
sin(x)
limx→0
x
x2 − 1
limx→1
x −1
x4
limx→∞ x .
e
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
5 / 11
Indeterminate products: “0 · ∞”
Example
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
6 / 11
Indeterminate products: “0 · ∞”
Example
Evaluate the limit: limx→0+ x ln(x).
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
6 / 11
Indeterminate products: “0 · ∞”
Example
Evaluate the limit: limx→0+ x ln(x).
You can see the difficulty. The first function, x, is trying to take the
product to 0; the second function, ln(x), is trying to take the product
to −∞.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
6 / 11
Indeterminate products: “0 · ∞”
Example
Evaluate the limit: limx→0+ x ln(x).
You can see the difficulty. The first function, x, is trying to take the
product to 0; the second function, ln(x), is trying to take the product
to −∞.
This is an indeterminate product of the type 0 · ∞. How can we
resolve this limit?
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
6 / 11
Indeterminate products: “0 · ∞”
Example
Evaluate the limit: limx→0+ x ln(x).
You can see the difficulty. The first function, x, is trying to take the
product to 0; the second function, ln(x), is trying to take the product
to −∞.
This is an indeterminate product of the type 0 · ∞. How can we
resolve this limit?
Strategy for the “0 · ∞” form
Suppose that the limit limx→a f (x)g (x) is of the form 0 · ∞. This form
can be converted into either a “0/0” or an “∞/∞” form by algebra:
f (x)g (x) =
g (x)
1/f (x)
or
f (x)g (x) =
f (x)
.
1/g (x)
Now the limit can be attacked by the previous methods.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
6 / 11
Indeterminate products: “0 · ∞”
Problem
Evaluate the following limits:
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
7 / 11
Indeterminate products: “0 · ∞”
Problem
Evaluate the following limits:
Find limx→0+ x ln(x).
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
7 / 11
Indeterminate products: “0 · ∞”
Problem
Evaluate the following limits:
Find limx→0+ x ln(x).
Find limx→∞ xe −x .
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
7 / 11
Indeterminate products: “0 · ∞”
Problem
Evaluate the following limits:
Find limx→0+ x ln(x).
Find limx→∞ xe −x .
Find limx→∞ x π/2 − tan−1 (x) .
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
7 / 11
Indeterminate differences: “∞ − ∞”
The basic strategy for indeterminate differences
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
8 / 11
Indeterminate differences: “∞ − ∞”
The basic strategy for indeterminate differences
An indeterminate difference is any limit of the form
lim f (x) − g (x)
x→a
in which f and g simultaneously approach +∞ or −∞.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
8 / 11
Indeterminate differences: “∞ − ∞”
The basic strategy for indeterminate differences
An indeterminate difference is any limit of the form
lim f (x) − g (x)
x→a
in which f and g simultaneously approach +∞ or −∞.
To handle an “∞ − ∞”’ form, use algebra to convert this form into
one of the other forms.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
8 / 11
Indeterminate differences: “∞ − ∞”
Problem
Find lim+
u→0
1
1
−
.
1 − e −u
u
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
9 / 11
Indeterminate differences: “∞ − ∞”
Problem
Find lim+
u→0
1
1
−
.
1 − e −u
u
Solution
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
9 / 11
Indeterminate differences: “∞ − ∞”
Problem
Find lim+
u→0
1
1
−
.
1 − e −u
u
Solution
This is called an “∞ − ∞” form, since both terms are approaching
+∞.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
9 / 11
Indeterminate differences: “∞ − ∞”
Problem
Find lim+
u→0
1
1
−
.
1 − e −u
u
Solution
This is called an “∞ − ∞” form, since both terms are approaching
+∞.
We can force a common denominator:
1
1
u − 1 + e −u
− =
.
−u
1−e
u
u(1 − e −u )
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
9 / 11
Indeterminate differences: “∞ − ∞”
Problem
Find lim+
u→0
1
1
−
.
1 − e −u
u
Solution
This is called an “∞ − ∞” form, since both terms are approaching
+∞.
We can force a common denominator:
1
1
u − 1 + e −u
− =
.
−u
1−e
u
u(1 − e −u )
As u → 0+ , the right-hand-side is now a “0/0” form and can be
treated using l’Hôpital’s rule. The answer is 1/2.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
9 / 11
Indeterminate powers: “00 ”, “∞0 ” and “1∞ ”
The basic strategy for indeterminate powers
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
10 / 11
Indeterminate powers: “00 ”, “∞0 ” and “1∞ ”
The basic strategy for indeterminate powers
An indeterminate power is any limit of the form
lim f (x)g (x)
x→a
resulting in “00 ”, “∞0 ” and “1∞ ”.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
10 / 11
Indeterminate powers: “00 ”, “∞0 ” and “1∞ ”
The basic strategy for indeterminate powers
An indeterminate power is any limit of the form
lim f (x)g (x)
x→a
resulting in “00 ”, “∞0 ” and “1∞ ”.
In each of these cases, first write
f (x)g (x) = exp g (x) ln f (x) .
The exponent, g (x) ln f (x), will be in one the preceding forms and
can be handled by those methods.
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
10 / 11
Indeterminate powers: “00 ”, “∞0 ” and “1∞ ”
Problem
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
11 / 11
Indeterminate powers: “00 ”, “∞0 ” and “1∞ ”
Problem
1 x
Find lim 1 + 2 .
x→∞
x
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
11 / 11
Indeterminate powers: “00 ”, “∞0 ” and “1∞ ”
Problem
1 x
Find lim 1 + 2 .
x→∞
x
Find limx→∞ x 1/x .
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
11 / 11
Indeterminate powers: “00 ”, “∞0 ” and “1∞ ”
Problem
1 x
Find lim 1 + 2 .
x→∞
x
Find limx→∞ x 1/x .
Find limx→0+ x sin(x) .
Mark Woodard (Furman U)
§7.8–Indeterminate forms and L’Hôpital’s rule
Fall 2010
11 / 11