1. Chapter 6
Differentiation

2. Section 6.3
L’Hospital’s Rule

3. Consider the limit where f (x)  L and g(x)  M.
If g(x)  0 for x close to c and M  0, then Theorem says that
If both L and M are zero, we can sometimes evaluate the limit by canceling a common
factor in the quotient.
In this section we derive another technique that is often easier
to use than factoring and has wider application.
In general, when L = M = 0, the limit of the quotient f /g is called an
indeterminate form, because different values may be obtained for the limit.
For example, for any real number k, let f (x) = kx and g(x) = x.
Then
So the indeterminate form (sometimes labeled 0/0) can lead to any real number
as the limit.

4. (Cauchy Mean Value Theorem) Theorem 6.3.1
Let f and g be functions that are continuous on [a, b] and differentiable on (a, b).
Then there exists at least one point c  (a, b) such that
[ f (b) – f (a)] g (c) = [g (b) – g (a)] f  (c).
Theorem 6.3.1
Proof:
Let h (x) = [ f (b) – f (a)]g (x) – [g (b) – g (a)] f (x) for each x  [a, b].
Then h is continuous on [a, b] and differentiable on (a, b).
Furthermore,
h (a) = [ f (b) – f (a)]g (a) – [g (b) – g (a)] f (a)
= f (b) g (a) – g (b) f (a)
h (b) = [ f (b) – f (a)]g (b) – [g (b) – g (a)] f (b)
= f (b) g (a) – g (b) f (a)
So, h(a) = h(b).
Thus, by the mean value theorem (or Rolle’s Theorem), there exists
c  (a, b) such that h(c) = 0.
That is, [ f (b) – f (a)] g (c) – [g (b) – g (a)] f  (c) = 0. 

5. Theorem 6.3.2 (L’Hospital’s Rule)
In the expression [ f (b) – f (a)] g (c) = [g (b) – g (a)] f  (c),
or,
if g(x) = x for all x, then we have
This formula is the original mean value theorem.
Theorem 6.3.2
(L’Hospital’s Rule)
Let f and g be continuous on [a, b] and differentiable on (a, b). Suppose that c  [a, b] and
that f (c) = g (c) = 0. Suppose also that g (x)  0 for x  U, where U is the intersection of
(a, b) and some deleted neighborhood of c. If
with L  , then

6. L L We have c  [a, b], f (c) = g (c) = 0, g (x)  0 for x  U, and
Proof:
Let (xn) be a sequence in U that converges to c.
Apply the Cauchy mean value theorem to f and g on the intervals [xn, c] or [c, xn]
to obtain a sequence (cn) with cn between xn and c for each n, such that
[ f (xn) – f (c)] g (cn) = [g (xn) – g (c)] f (cn).
Since g (x)  0 for all x  U, and g (c) = 0, we must have g (xn)  0 for all n.
(See Rolle’s Theorem.)
Thus, since f (c) = g(c) = 0, we have L L
for all n.
Furthermore, since xn  c and cn is between xn and c, it follows that cn  c.
Thus by Theorem 5.1.8,
But then,
so , also. 

7. Example 6.3.3 Example 6.3.4 Let f (x) = 2x2 – 3x + 1 and g(x) = x – 1.
Then f (1) = g(1) = 0.
And we have f (x) = 4x – 3 and g (x) = 1, so that
We did this same problem by
factoring in Example
Example 6.3.4
Let f (x) = 1 – cos x and g(x) = x2.
Then f (0) = g(0) = 0.
And we have f (x) = sin x and g (x) = 2x, so that
Since sin x  0 and 2x  0 as x  0,
we again have the indeterminate form 0/0.
provided that the second limit exists.
So we use l’Hospital’s rule again.
Note that g (x) must be nonzero in a deleted neighborhood of 0, but it is permitted
that g (0) = 0.

8. Definition 6.3.6 Definition 6.3.7
In some situations we wish to evaluate the limit of a function for larger and larger
values of the variable.
Definition 6.3.6
Let f : (a, )  We say that the real number L is the limit of f as x  , and we
write provided that for each  > 0 there exists a real number N > a
such that x > N implies that |  f (x) – L | < .
Very often as x   the values of a given function also get large. This leads to the
following definition.
Definition 6.3.7
Let f : (a, )  We say that f tends to  as x  , and we write
provided that given any   there exists an N > a such that x > N implies that f (x) >  .
Using these definitions, we can state l’Hospital’s rule for indeterminates of the form /.

9. Theorem 6.3.8 (L’Hospital’s Rule) with L  , then
Let f and g be differentiable on (a, ). Suppose that limx    f (x) = limx   g (x) = ,
and that g (x)  0 for x  (a, ). If
with L  , then
The proof is lengthy, but basically consists of the Cauchy mean value theorem and algebra.
There is also an extension of l’Hospital’s rule that applies to f /g when
lim x  c f (x) = lim x  c g(x) =  and c  (See Exercise 13.)
There are other limiting situations involving two functions that can give rise to
ambiguous values. These indeterminate forms are indicated by the symbols
0  , 00, 1,  0, and  – , and are evaluated by using algebraic manipulations,
logarithms, or exponentials to change them into one of the forms 0/0 or /.

10. Example
For x > 0, let f (x) = x and g(x) = – ln x. Then lim x  0+ f (x)g(x) is an indeterminate
of the form 0  .
To evaluate the limit, we write 
= 0. 
From this we also conclude that lim x  0+ x ln x = 0.