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]

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(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]

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[pic]

Thus, [pic][pic]

[pic]

(25) Let [pic]

[pic]. Thus, [pic]

[pic][pic].

(26) [pic]

[pic].

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