Limits – Indeterminate Forms and L’Hospital’s Rule
Limits – Indeterminate Forms and L’Hopital’s Rule
I. Indeterminate Form of the Type [pic]
We have previously studied limits with the indeterminate form [pic] as shown in the
following examples:
Example 1: [pic]
Example 2: [pic]
[pic]
[Note: We use the given limit [pic].]
Example 3: [pic]. [Note: We use the definition
of the derivative [pic] where [pic]
and a = 8.]
Example 4: [pic]. [Note: We use the
definition of the derivative [pic] where
[pic] and [pic].]
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However, there is a general, systematic method for determining limits with the
indeterminate form [pic]. Suppose that f and g are differentiable functions at x = a
and that [pic] is an indeterminate form of the type [pic]; that is, [pic]
and [pic]. Since f and g are differentiable functions at x = a, then f and g
are continuous at x = a; that is, [pic]= 0 and [pic]= 0.
Furthermore, since f and g are differentiable functions at x = a, then [pic]
[pic] and [pic]. Thus, if [pic], then
[pic] if[pic] and
[pic] are continuous at x = a. This illustrates a special case of the technique known as
L’Hopital’s Rule.
L’Hopital’s Rule for Form [pic]
Suppose that f and g are differentiable functions on an open interval
containing x = a, except possibly at x = a, and that [pic] and
[pic]. If [pic] has a finite limit, or if this limit is [pic] or
[pic], then [pic]. Moreover, this statement is also true
in the case of a limit as [pic]or as [pic]
In the following examples, we will use the following three-step process:
Step 1. Check that the limit of [pic] is an indeterminate form of type [pic]. If it
is not, then L’Hopital’s Rule cannot be used.
Step 2. Differentiate f and g separately. [Note: Do not differentiate [pic]
using the quotient rule!]
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Step 3. Find the limit of [pic]. If this limit is finite, [pic], or [pic], then it is
equal to the limit of [pic]. If the limit is an indeterminate form of type
[pic], then simplify [pic] algebraically and apply L’Hopital’s Rule again.
Example 1: [pic]
Example 2: [pic]
Example 3: [pic]
Example 4: [pic]
Example 5: [pic] [Use L’Hopital’s Rule
twice.]
Example 6: [pic], or
[pic] where [pic].
Example 7: [pic][This limit is not an indeterminate
form of the type [pic], so L’Hopital’s Rule cannot be used.]
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II. Indeterminate Form of the Type [pic]
We have previously studied limits with the indeterminate form [pic] as shown in the
following examples:
Example 1: [pic]
[pic]
Example 2: [pic] [pic] [pic]
Example 3: [pic] [pic] [pic]
limit does not exist.
Example 4: [pic] [pic] [pic][pic]
because x < 0 and thus [pic]) = [pic]
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[pic].
However, we could use another version of L’Hopital’s Rule.
L’Hopital’s Rule for Form [pic]
Suppose that f and g are differentiable functions on an open interval
containing x = a, except possibly at x = a, and that [pic] and
[pic]. If [pic] has a finite limit, or if this limit is [pic] or
[pic], then [pic]. Moreover, this statement is also true
in the case of a limit as [pic]or as [pic]
Example 1: [pic]
Example 2: [pic]
Example 3: [pic]
Example 4: [pic] L’Hopital’s
Rule does not help in this situation. We would find the limit as we
did previously.
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Example 5: [pic]=
[pic]
Example 6: [pic]
Example 7: [pic] [This limit is
not an indeterminate form of the type [pic], so L’Hopital’s Rule
cannot be used.]
III. Indeterminate Form of the Type [pic]
Indeterminate forms of the type [pic] can sometimes be evaluated by rewriting the
product as a quotient, and then applying L’Hopital’s Rule for the indeterminate
forms of type [pic] or [pic].
Example 1: [pic]
Example 2: [pic]
[pic]
Example 3: [pic] [Let [pic].]
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IV. Indeterminate Form of the Type [pic]
A limit problem that leads to one of the expressions
[pic], [pic], [pic], [pic]
is called an indeterminate form of type [pic]. Such limits are indeterminate
because the two terms exert conflicting influences on the expression; one pushes
it in the positive direction and the other pushes it in the negative direction. However,
limits problems that lead to one the expressions
[pic], [pic], [pic], [pic]
are not indeterminate, since the two terms work together (the first two produce a
limit of [pic] and the last two produce a limit of [pic]). Indeterminate forms of the
type [pic] can sometimes be evaluated by combining the terms and manipulating
the result to produce an indeterminate form of type [pic] or [pic].
Example 1: [pic]
[pic]
Example 2: [pic]
[pic]
V. Indeterminate Forms of the Types [pic]
Limits of the form [pic] [pic] frequently give rise to
indeterminate forms of the types [pic]. These indeterminate forms can
sometimes be evaluated as follows:
1) [pic]
2) [pic]
3) [pic]
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The limit on the righthand side of the equation will usually be an
indeterminate limit of the type [pic]. Evaluate this limit using the
technique previously described. Assume that [pic]= L.
(4) Finally, [pic].
Example 1: Find [pic].
This is an indeterminate form of the type [pic]. Let [pic]
[pic]. [pic]0.
Thus, [pic].
Example 2: Find [pic].
This is an indeterminate form of the type [pic]. Let [pic]
[pic]. [pic]=
[pic]. Thus, [pic]=
[pic].
Example 3: Find [pic].
This is an indeterminate form of the type [pic]. Let [pic]
[pic]. [pic]
[pic]. Thus, [pic] = [pic].
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Practice Sheet for L’Hopital’s Rule
(1) [pic]
(2) [pic]
(3) [pic]
(4) [pic]
(5) [pic]
(6) [pic]
(7) [pic]
(8) [pic]
(9) [pic]
(10) [pic]
(11) [pic]
9
(12) [pic]
(13) [pic]
(14) [pic]
(15) [pic]
(16) [pic]
(17) [pic]
(18) [pic]
(19) [pic]
(20) [pic]
(21)[pic]
(22) [pic]
(23) [pic]
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(24) [pic]
(25) [pic]
(26) [pic]
Solution Key for L’Hopital’s Rule
(1) [pic]
(2) [pic]
[pic]
(3) [pic]
[pic]
(4) Let [pic][pic]. Now, let [pic]
[pic]. Thus,
[pic][pic] =
[pic].
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(5) Let [pic][pic]
[pic].
(6) [pic]
[pic].
(7) Let [pic]
[pic]. Thus, [pic]
[pic][pic].
(8) [pic]
[pic] limit does not exist.
(9) [pic]
[pic].
(10) [pic]
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(11) Let [pic]
[pic]. Thus, [pic]
[pic][pic].
(12) [pic]
[pic].
(13) [pic]
[pic].
(14) [pic]
(15) [pic]
(16) [pic].
(17) Let [pic][pic]. Next, let [pic]
[pic]. Thus,
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[pic] [pic] =
[pic].[pic]
(18) [pic].
(19) [pic].
(20) Let [pic]
[pic]. Thus [pic]
[pic][pic].
(21) [pic].
(22) [pic].
(23) [pic].
(24) Let [pic][pic] Let [pic]
14
[pic]
Thus, [pic][pic]
[pic]
(25) Let [pic]
[pic]. Thus, [pic]
[pic][pic].
(26) [pic]
[pic].
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