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In this section, we will learn:
How to evaluate functions whose values cannot be found at certain points using derivatives.
1.
4
APPLICATIONS OF DIFFERENTIATION
2.
APPLICATIONS OF DIFFERENTIATION
4.4
Indeterminate Forms
and L’Hospital’s Rule
In this section, we will learn:
How to evaluate functions whose
values cannot be found at certain points.
3.
INDETERMINATE FORMS
Suppose we are trying to analyze
the behavior of the function ln x
F ( x)
x 1
Although F is not defined when x = 1,
we need to know how F behaves near 1.
4.
INDETERMINATE FORMS Expression 1
In particular, we would like to know
the value of the limit
ln x
lim
x 1 x 1
5.
INDETERMINATE FORMS
In computing this limit, we can’t apply
Law 5 of limits (Section 2.3) because
the limit of the denominator is 0.
In fact, although the limit in Expression 1 exists,
its value is not obvious because both numerator
0
and denominator approach 0 and is not defined.
0
6.
INDETERMINATE FORM —TYPE 0/0
In general, if we have a limit of the form f ( x)
lim
x a g ( x)
where both f(x) → 0 and g(x) → 0 as x → a,
then this limit may or may not exist.
0
It is called an indeterminate form of type .
0
We met some limits of this type in Chapter 2.
7.
INDETERMINATE FORMS
For rational functions, we can cancel
common factors:
2
x x x( x 1)
lim 2 lim
x 1 x 1 x 1 ( x 1)( x 1)
x 1
lim
x 1 x 1 2
8.
INDETERMINATE FORMS
We used a geometric argument
to show that:
sin x
lim 1
x 0 x
9.
INDETERMINATE FORMS
However, these methods do not work
for limits such as Expression 1.
Hence, in this section, we introduce
a systematic method, known as l’Hospital’s Rule,
for the evaluation of indeterminate forms.
10.
INDETERMINATE FORMS Expression 2
Another situation in which a limit is
not obvious occurs when we look for
a horizontal asymptote of F and need
to evaluate the limit ln x
lim
x x 1
11.
INDETERMINATE FORMS
It isn’t obvious how to evaluate this limit
because both numerator and denominator
become large as x → ∞.
There is a struggle between the two.
If the numerator wins, the limit will be ∞.
If the denominator wins, the answer will be 0.
Alternatively, there may be some compromise—
the answer may be some finite positive number.
12.
INDETERMINATE FORM —TYPE ∞/∞
In general, if we have a limit of the form
f ( x)
lim
x a g ( x)
where both f(x) → ∞ (or -∞) and g(x) → ∞
(or -∞), then the limit may or may not exist.
It is called an indeterminate form of type ∞/∞.
13.
INDETERMINATE FORMS
We saw in Section 2.6 that this type of limit
can be evaluated for certain functions—
including rational functions—by dividing the
numerator and denominator by the highest
power of x that occurs in the denominator.
For instance, 1
2 1 2
x 1 x 1 0 1
lim 2 lim
x 2 x 1 x 1 20 2
2 2
x
14.
INDETERMINATE FORMS
This method, though, does not work
for limits such as Expression 2.
However, L’Hospital’s Rule also applies to
this type of indeterminate form.
15.
L’HOSPITAL’S RULE
Suppose f and g are differentiable and
g’(x) ≠ 0 on an open interval I that contains a
(except possibly at a).
Suppose lim
x a
f ( x) 0 and lim g ( x) 0
x a
or that lim f ( x) and lim g ( x)
x a x a
In other words, we have an indeterminate form
0
of type or ∞/∞.
0
16.
L’HOSPITAL’S RULE
f ( x) f '( x)
lim lim
x a g ( x) x a g '( x )
if the limit on the right exists
(or is ∞ or - ∞).
17.
NOTE 1
L’Hospital’s Rule says that the limit of
a quotient of functions is equal to the limit of
the quotient of their derivatives—provided that
the given conditions are satisfied.
It is especially important to verify the conditions
regarding the limits of f and g before using the rule.
18.
NOTE 2
The rule is also valid for one-sided limits
and for limits at infinity or negative infinity.
That is, ‘x → a’ can be replaced by any of
the symbols x → a+, x → a-, x → ∞, or x → - ∞.
19.
NOTE 3
For the special case in which
f(a) = g(a) = 0, f’ and g’ are continuous,
and g’(a) ≠ 0, it is easy to see why
the rule is true.
20.
NOTE 3
In fact, using the alternative form of
the definition of a derivative, we have:
f ( x) f (a) f ( x) f (a)
lim
f '( x) f '(a ) x a x a x a
lim lim
x a g '( x ) g '(a ) g ( x) g (a ) x a g ( x) g (a )
lim
x a x a x a
f ( x) f (a)
lim
x a g ( x) g (a)
f ( x)
lim
x a g ( x)
21.
NOTE 3
It is more difficult to prove
the general version of l’Hospital’s
22.
L’HOSPITAL’S RULE Example 1
ln x
Find lim
x 1 x 1
lim ln x ln1 0 and lim( x 1) 0
x 1 x 1
Thus, we can apply l’Hospital’s Rule:
d
(ln x)
ln x dx 1/ x 1
lim lim lim lim 1
x 1 x 1 x 1 d x 1 1 x 1 x
( x 1)
dx
23.
L’HOSPITAL’S RULE Example 2
x
e
Calculate lim 2
x x
We have lim e x and lim x 2
x x
So, l’Hospital’s Rule gives:
d x
x (e ) x
e e
lim 2 lim dx lim
x x x d 2 x 2 x
(x )
dx
24.
L’HOSPITAL’S RULE Example 2
As ex → ∞ and 2x → ∞ as x → ∞, the limit
on the right side is also indeterminate.
However, a second application of l’Hospital’s
Rule gives: ex ex ex
lim 2 lim lim
x x x 2 x x 2
25.
L’HOSPITAL’S RULE Example 3
ln x
Calculate lim 3
x x
As ln x → ∞ and 3 x as x → ∞,
l’Hospital’s Rule applies: ln x 1/ x
lim lim 1 2 / 3
x 3 x x 3 x
Notice that the limit on the right side
0
is now indeterminate of type .
0
26.
L’HOSPITAL’S RULE Example 3
However, instead of applying the rule
a second time as we did in Example 2,
we simplify the expression and see that
a second application is unnecessary:
ln x 1/ x 3
lim lim 1 2 / 3 lim 3 0
x 3
x x 3 x x
x
27.
L’HOSPITAL’S RULE Example 4
tan x x
Find lim 3
x 0 x
Noting that both tan x – x → 0 and x3 → 0
as x → 0, we use l’Hospital’s Rule:
2
tan x x sec x 1
lim 3
lim
x 0 x x 0 3x 2
28.
L’HOSPITAL’S RULE Example 4
As the limit on the right side is still
0
indeterminate of type , we apply the rule
0
again:
2 2
sec x 1 2sec x tan x
lim 2
lim
x 0 3x x 0 6x
29.
L’HOSPITAL’S RULE Example 4
2
Since lim sec x 1 , we simplify the
x 0
calculation by writing:
2
2sec x tan x 1 2 tan x
lim lim sec x lim
x 0 6x 3 x 0 x 0 x
1 tan x
lim
3 x 0 x
30.
L’HOSPITAL’S RULE Example 4
We can evaluate this last limit either by
using l’Hospital’s Rule a third time or by
writing tan x as (sin x)/(cos x) and making
use of our knowledge of trigonometric limits.
31.
L’HOSPITAL’S RULE Example 4
Putting together all the steps,
we get:
2
tan x x sec x 1
lim 3
lim 2
x 0 x x 0 3x
2
2sec x tan x
lim
x 0 6x
2
1 tan x 1 sec x 1
lim lim
3 x 0 x 3 x 0 1 3
32.
L’HOSPITAL’S RULE Example 5
sin x
Find xlim
1 cos x
If we blindly attempted to use l-Hospital’s rule,
we would get: sin x cos x
lim lim
x 1 cos x x sin x
33.
L’HOSPITAL’S RULE Example 5
This is wrong.
Although the numerator sin x → 0 as x → π -,
notice that the denominator (1 - cos x) does not
approach 0.
So, the rule can’t be applied here.
34.
L’HOSPITAL’S RULE Example 5
The required limit is, in fact, easy to find
because the function is continuous at π
and the denominator is nonzero there:
sin x sin 0
lim 0
x 1 cos x 1 cos 1 ( 1)
35.
L’HOSPITAL’S RULE
The example shows what can go wrong
if you use the rule without thinking.
Other limits can be found using the rule, but
are more easily found by other methods.
See Examples 3 and 5 in Section 2.3,
Example 3 in Section 2.6, and the discussion
at the beginning of the section.
36.
L’HOSPITAL’S RULE
So, when evaluating any limit,
you should consider other methods
before using l’Hospital’s Rule.
37.
INDETERMINATE PRODUCTS
If lim f ( x) 0 and lim g ( x) (or -∞),
x a x a
then it isn’t clear what the value
of lim f ( x) g ( x), if any, will be.
x a
38.
INDETERMINATE PRODUCTS
There is a struggle between f and g.
If f wins, the answer will be 0.
If g wins, the answer will be ∞ (or -∞).
Alternatively, there may be a compromise
where the answer is a finite nonzero number.
39.
INDETERMINATE FORM—TYPE 0 . ∞
This kind of limit is called an indeterminate
form of type 0 . ∞.
We can deal with it by writing the product fg
as a quotient: f g
fg or fg
1/ g 1/ f
This converts the given limit into an indeterminate form
0
of type or ∞/∞, so that we can use l’Hospital’s Rule.
0
40.
INDETERMINATE PRODUCTS Example 6
Evaluate lim x ln x
x 0
The given limit is indeterminate because,
as x → 0+, the first factor (x) approaches 0,
whereas the second factor (ln x) approaches -∞.
41.
INDETERMINATE PRODUCTS Example 6
Writing x = 1/(1/x), we have 1/x → ∞
as x → 0+.
So, l’Hospital’s Rule gives:
ln x 1/ x
lim x ln x lim lim 2
x 0 x 0 1/ x x 0 1/ x
lim ( x) 0
x 0
42.
INDETERMINATE PRODUCTS Note
In solving the example, another possible
option would have been to write:
x
lim x ln x lim
x 0 x 0 1/ ln x
This gives an indeterminate form of the type 0/0.
However, if we apply l’Hospital’s Rule, we get a more
complicated expression than the one we started with.
43.
INDETERMINATE PRODUCTS Note
In general, when we rewrite an
indeterminate product, we try to choose
the option that leads to the simpler limit.
44.
INDETERMINATE FORM—TYPE ∞ -∞
If lim f ( x) and lim g ( x) , then
x a x a
the limit
lim[ f ( x) g ( x)]
x a
is called an indeterminate form
of type ∞ - ∞.
45.
INDETERMINATE DIFFERENCES
Again, there is a contest between f and g.
Will the answer be ∞ (f wins)?
Will it be - ∞ (g wins)?
Will they compromise on a finite number?
46.
INDETERMINATE DIFFERENCES
To find out, we try to convert the difference
into a quotient (for instance, by using a
common denominator, rationalization, or
factoring out a common factor) so that
0
we have an indeterminate form of type
0
or ∞/∞.
47.
INDETERMINATE DIFFERENCES Example 7
Compute lim (sec x tan x)
x ( / 2)
First, notice that sec x → ∞ and tan x → ∞
as x → (π/2)-.
So, the limit is indeterminate.
48.
INDETERMINATE DIFFERENCES Example 7
Here, we use a common denominator:
1 sin x
lim (sec x tan x) lim
x ( / 2) x ( / 2) cos x cos x
1 sin x
lim
x ( / 2) cos x
cos x
lim 0
x ( / 2) sin x
Note that the use of l’Hospital’s Rule is justified
because 1 – sin x → 0 and cos x → 0 as x → (π/2)-.
49.
INDETERMINATE POWERS
Several indeterminate forms arise from
g ( x)
the limit lim[ f ( x)]
x a
0
1. lim f ( x) 0 and lim g ( x) 0 type 0
x a x a
0
2. lim f ( x) and lim g ( x) 0 type
x a x a
3. lim f ( x) 1 and lim g ( x) type1
x a x a
50.
INDETERMINATE POWERS
Each of these three cases can be treated
in either of two ways.
Taking the natural logarithm:
Let y [ f ( x)]g ( x ) , then ln y g ( x) ln f ( x)
Writing the function as an exponential:
g ( x) g ( x )ln f ( x )
[ f ( x)] e
51.
INDETERMINATE POWERS
Recall that both these methods were used
in differentiating such functions.
In either method, we are led to the indeterminate
product g(x) ln f(x), which is of type 0 . ∞.
52.
INDETERMINATE POWERS Example 8
cot x
Calculate lim (1 sin 4 x)
x 0
First, notice that, as x → 0+, we have
1 + sin 4x → 1 and cot x → ∞.
So, the given limit is indeterminate.
53.
INDETERMINATE POWERS Example 8
Let y = (1 + sin 4x)cot x
Then, ln y = ln[(1 + sin 4x)cot x]
= cot x ln(1 + sin 4x)
54.
INDETERMINATE POWERS Example 8
So, l’Hospital’s Rule gives:
ln(1 sin 4 x)
lim ln y lim
x 0 x 0 tan x
4 cos 4 x
lim 1 sin2 4 x 4
x 0 sec x
55.
INDETERMINATE POWERS Example 8
So far, we have computed the limit of ln y.
However, what we want is the limit of y.
To find this, we use the fact that y = eln y:
cot x
lim (1 sin 4 x) lim y
x 0 x 0
lim eln y e 4
x 0
56.
INDETERMINATE POWERS Example 9
x
Find lim x
x 0
Notice that this limit is indeterminate
since 0x = 0 for any x > 0 but x0 = 1
for any x ≠ 0.
57.
INDETERMINATE POWERS Example 9
We could proceed as in Example 8 or by
writing the function as an exponential:
xx = (eln x)x = ex ln x
In Example 6, we used l’Hospital’s Rule
to show that lim x ln x 0
x 0
x x ln x 0
Therefore, lim x lim e e 1
x 0 x 0
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