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Finding Limits of Compositions

Instructor: Christopher Haines
In this lesson, we learn how to evaluate a limit of a composition of two functions. Under certain conditions, we have a nice formula for this limit. If the conditions in the theorem are violated, we must resort to other methods of evaluating the limit.

The Description of the Problem

Suppose we have two functions f and g having limits


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and


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where a, b and c are any extended real numbers (possibly positive/negative infinity).

Based on these two limits of f and g, we ask whether the following limit exists, and if so, can we find it?


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As we will see, under considerable general conditions on the functions f and g, it will follow that


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For instance, consider the limit


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We recognize the function (within the limit) as a composition of the two functions


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Further,


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and


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By comparison with i., ii. and iii., a is infinity, b = 0, and c = 1. If f and g satisfied these conditions of this theorem, then it would follow that


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The next section is devoted to stating the main theorem of this lesson, while the third section shows several examples of situations in which the theorem does apply and when it does not apply. When the theorem does not apply, we must use another method to evaluate the limit of the composition.

The Limit of Compositions Theorem

Although our primary objective in this lesson is to evaluate a limit of a composition, we exercise some precaution in using the formula iii. The limit of compositions theorem serves the purpose of ensuring we do not incorrectly conclude iii. We now state this theorem.

Theorem: Limit of Compositions

Suppose i. and ii. hold and additionally, at least one of the following two conditions holds.


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Then


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We negate the proof of this theorem as our main purpose is to illustrate its use. Before continuing to examples, there is one key observation on Condition (1) in the theorem. Condition (1) together with ii. means that f is continuous at b. Further the conclusion in the theorem (by transitivity of equality) asserts


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This is an alternate way of writing iii. in the event that f is continuous at the point x = b. It has a more meaningful interpretation than the previous version, and that is the limit operator can be passed through the outer function. For most learners, this is the easiest of the two versions to remember. However, even when the conclusion of the theorem follows, it is possible for iii.' to be false. We will see this in one of the upcoming examples. We now give four examples in which the limit of compositions theorem can apply.

Examples of Limits of Compositions

Example 1: Opening Revisited

Evaluate the following limit.


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Solution

Recall that we confirmed the validity of i. and ii. with


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Now since f is continuous at x = 0, the limit of compositions theorem can be applied to conclude that


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Example 2: A Second Case Where f Is Continuous at x = b

Evaluate the limit


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Solution

Note that we can rewrite the limit as


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The outer function is


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and the inner function is


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In this example, we will need L'Hospital's Rule, and for that reason, we remind the reader of that theorem.

Theorem: L'Hospital's Rule

Suppose we have the following conditions on the limits of f and g at x = a.


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or both of these limits are infinite (positive or negative infinity).


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Then,


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Using L'Hospital's Rule, we see that


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and clearly f is continuous at x = 0. Thus, by the limit of compositions theorem,


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Example 3: f is Not Continuous at x = b

Evaluate the limit


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where


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and


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Solution

To begin, note that


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and that f is not continuous at x = 0. However g(x) > 0 for all x > 0, and hence condition 2 in the limit of compositions theorem holds. Therefore, we can conclude that


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Alternatively, we can evaluate the limit directly as follows.


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Also observe that in this example.


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Example 4: g(x) Equals b for Infinitely Many x Near a

Evaluate the limit


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Solution

In this example, we will need to use the Squeeze Theorem, and for that reason, we state this here.

Squeeze Theorem


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and


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Then,


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Since


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it follows from the Squeeze Theorem that


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To correspond with the limit of compositions theorem, let


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Since f is continuous at x = 0, the conclusion in the theorem follows, and


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It is also worth noting that Condition 2 in the theorem fails. This is because


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Essentially as g approaches its limit of zero from the right, it touches zero on a sequence of points defined by


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It is the same story as g approaches its limit of zero from the left. Every interval around zero will contain a part of that sequence (and an infinite number of terms in the sequence).


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