Section 4.4
Indeterminate Forms
and L’Hospital’s Rule
Applications of Differentiation
Suppose we are trying to analyze the behavior of the function
Although F is not defined when x = 1, we need to know how F behaves near 1.
ln( )
1
xF x
x
Indeterminate Forms
In particular, we would like to know the value of the limit
1
lnlim
1x
x
x Expression 1
Indeterminate Forms
In computing this limit, we can not 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 and denominator approach 0 and is not defined.0
0
Indeterminate Forms
In general, if we have a limit of the form
where both f(x) → 0 and g(x) → 0 as x → a, then this limit may or may not exist.
It is called an indeterminate form of type .
• We met some limits of this type in Chapter 2.
( )lim
( )x a
f x
g x
0
0
Indeterminate Form Type 0/0
For rational functions, we can cancel common factors:
2
21 1
1
( 1)lim lim
1 ( 1)( 1)
1lim
1 2
x x
x
x x x x
x x x
x
x
Indeterminate Forms
We can use a geometric argument to show that:
0
sinlim 1x
x
x
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.
Indeterminate Forms
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
lnlim
1x
x
x
Indeterminate Forms
Expression 2
It is not 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.
Indeterminate Forms
In general, if we have a limit of the form
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 ∞/∞.
( )lim
( )x a
f x
g x
Indeterminate Form Type ∞/∞
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,2 2
2
2
111 1 0 1
lim lim12 1 2 0 22
x x
x xx
x
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.
Indeterminate Forms
lnlim
1x
x
x
Suppose f and g are differentiable and g’(x) ≠ 0 on an open interval I that contains a (except possibly at a).
Suppose
or that
• In other words, we have an indeterminate form of type or ∞/∞.
lim ( ) 0 and lim ( ) 0x a x a
f x g x
0
0
lim ( ) and lim ( )x a x a
f x g x
L’Hopital’s Rule
Then,
provided that the limit on the right exists (or is ∞ or - ∞).
( ) '( )lim lim
( ) '( )x a x a
f x f x
g x g x
L’Hopital’s Rule
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.
Remark 1
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 → - ∞.
Remark 2
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.
Remark 3
In fact, using the alternative form of the definition of a derivative, we have:
( ) ( )lim'( ) '( )
lim( ) ( )'( ) '( ) lim
( ) ( )( ) ( ) ( )
lim lim lim( ) ( ) ( ) ( ) ( )
x a
x a
x a
x a x a x a
f x f af x f a x a
g x g ag x g ax a
f x f af x f a f xx a
g x g a g x g a g xx a
Remark 3
It is more difficult to prove the general version of l’Hospital’s Rule.
Remark 3
Find
• and
• Thus, we can apply l’Hospital’s Rule:
1
lnlim
1x
x
x
1 1lim ln ln1 0 lim( 1) 0x x
x x
1 1 1 1
(ln )ln 1/ 1lim lim lim lim 1
1 1( 1)x x x x
dxx xdx
dx xxdx
L’Hopital’s Rule - Example 1
Calculate
• We have and
• So, l’Hospital’s Rule gives:
2lim
x
x
e
x
2lim limx
x xe x
22
( )lim lim lim
2( )
xx x
x x x
dee edx
dx xxdx
L’Hopital’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:
2lim lim lim
2 2
x x x
x x x
e e e
x x
L’Hopital’s Rule - Example 2
Calculate
• As ln x → ∞ and as x → ∞, l’Hospital’s Rule applies:
• Notice that the limit on the right side is now indeterminate of type .
3
lnlimx
x
x
3 x
2/3133
ln 1/lim limx x
x x
xx
0
0
L’Hopital’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:
2/313 33
ln 1/ 3lim lim lim 0x x x
x x
xx x
L’Hopital’s Rule - Example 3
Find
• Noting that both tan x – x → 0 and x3 → 0 as x → 0, we use l’Hospital’s Rule:
30
tanlimx
x x
x
2
3 20 0
tan sec 1lim lim
3x x
x x x
x x
L’Hopital’s Rule - Example 4
• As the limit on the right side is still indeterminate of type , we apply the rule again:
2 2
20 0
sec 1 2sec tanlim lim
3 6x x
x x x
x x
0
0
L’Hopital’s Rule - Example 4
• Since , we simplify the calculation by writing:2
0limsec 1x
x
22
0 0 0
0
2sec tan 1 tanlim limsec lim
6 31 tan
lim3
x x x
x
x x xx
x xx
x
L’Hopital’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.
L’Hopital’s Rule - Example 4
2
3 20 0
2
0
2
0 0
tan sec 1lim lim
3
2sec tanlim
6
1 tan sec 1lim lim
3 1 3
x x
x
x x
x x x
x x
x x
x
x x
x
Find
• If we blindly attempted to use l-Hospital’s rule, we would get:
sinlim
1 cosx
x
x
sin coslim lim
1 cos sinx x
x x
x x
L’Hopital’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 not be applied here.
L’Hopital’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 sin 0lim 0
1 cos 1 cos 1 ( 1)x
x
x
L’Hopital’s Rule - Example 5
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.
So, when evaluating any limit, you should consider other methods before using l’Hospital’s Rule.
L’Hopital’s Rule
If and (or -∞),
then it is not clear what the value of if any, will be.
lim ( ) 0x a
f x
lim ( )x a
g x
lim ( ) ( )x a
f x g x
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.
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:
• This converts the given limit into an indeterminate form of type or ∞/∞, so that we can use l’Hospital’s Rule.
or1/ 1/
f gfg fg
g f
0
0
Indeterminate Form Type 0
. ∞
Evaluate
• The given limit is indeterminate because, as x → 0+, the first factor (x) approaches 0, whereas the second factor (ln x) approaches -∞.
0lim lnx
x x
Indeterminate Products – Ex 6
• Writing x = 1/(1/x), we have 1/x → ∞ as
x → 0+.
• So, l’Hospital’s Rule gives:
20 0 0
0
ln 1/lim ln lim lim
1/ 1/lim ( ) 0
x x x
x
x xx x
x xx
Indeterminate Products – Ex 6
In solving the example, another possible option would have been to write:
• 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.
• In general, when we rewrite an indeterminate product, we choose the option that leads to the simpler limit.
0 0lim ln lim
1/ lnx x
xx x
x
Indeterminate Products – Remark
If and , then
is called an indeterminate form of type ∞ - ∞.
lim ( )x a
f x
lim ( )x a
g x
lim[ ( ) ( )]x a
f x g x
Indeterminate Form Type ∞ – ∞
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?
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 we have an indeterminate form of type or ∞/∞.0
0
Indeterminate Differences
Compute
• First, notice that sec x → ∞ and tan x → ∞ as x → (π/2)-.
• So, the limit is indeterminate.
( / 2)lim (sec tan )
xx x
Indeterminate Differences – Ex 7
Indeterminate Differences – Ex 7 Here, we use a common denominator:
• Note that the use of l’Hospital’s Rule is justified because
1 – sin x → 0 and cos x → 0 as x → (π/2)-.
( / 2) ( / 2)
( / 2)
( / 2)
1 sinlim (sec tan ) lim
cos cos
1 sinlim
coscos
lim 0sin
x x
x
x
xx x
x x
x
xx
x
Several indeterminate forms arise from( )lim[ ( )]g x
x af x
0
0
1. lim ( ) 0 and lim ( ) 0 type0
2. lim ( ) and lim ( ) 0 type
3. lim ( ) 1 and lim ( ) type1
x a x a
x a x a
x a x a
f x g x
f x g x
f x g x
Indeterminate Powers
Each of these three cases can be treated in either of two ways.
• Taking the natural logarithm:
• Writing the function as an exponential:
• In either method, we are led to the indeterminate product
g(x) ln f(x), which is of type 0·∞.
( )Let [ ( )] , then ln ( ) ln ( )g xy f x y g x f x
( ) ( ) ln ( )[ ( )]g x g x f xf x e
Indeterminate Powers
Calculate
• First, notice that, as x → 0+, we have 1 + sin 4x → 1 and
cot x → ∞.
• So, the given limit is indeterminate.
cot
0lim (1 sin 4 ) x
xx
Indeterminate Powers – Ex 8
Let y = (1 + sin 4x)cot x
Then, ln y = ln[(1 + sin 4x)cot x] = cot x ln(1 + sin 4x)
Indeterminate Powers – Ex 8
So, l’Hospital’s Rule gives:
0 0
20
ln(1 sin 4 )lim ln lim
tan4cos 4
1 sin 4lim 4sec
x x
x
xy
xxx
x
Indeterminate Powers – Ex 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
0 0
ln 4
0
lim(1 sin 4 ) lim
lim
x
x x
y
x
x y
e e
Indeterminate Powers – Ex 8
Find
• Notice that this limit is indeterminate since 0x = 0 for any
x > 0 but x0 = 1 for any x ≠ 0.
0lim x
xx
Indeterminate Powers – Ex 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
• Therefore,
0lim ln 0x
x x
ln 0
0 0lim lim 1x x x
x xx e e
Indeterminate Powers – Ex 9
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