Why Are Assumptions Critical in the Limit of Composite Functions?

In summary: The assumptions are necessary in order to prove the theorem, but they may not be necessary for the statement of the theorem to be true.
  • #1
sonofagun
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I need help with the following theorem:

Let I, J ⊆ℝ be open intervals, let x∈I, let g: I\{x}→ℝ and f: J→ℝ be functions with g[I\{x}]⊆J and Limz→xg(x)=L∈J. Assume that limy→L f(y) exists and that g[I\{x}]⊆J\{g(x)},or, in case g(x)∈g[I\{x}] that limy→L f(y)=f(L). Then f(g(x)) converges at x, and limz→x f(g(x))=limy→Lf(y).

I don't get why the assumptions are necessary. We're assuming that the limit of f exists when g is not defined at x, or, if g is defined at x, that the limy→L f (y)=f(L).
 
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  • #2
sonofagun said:
I don't get why the assumptions are necessary.

What assumptions would you propose to use instead?

Do you mean "necessary" in the mathematical sense (as in "necessary and sufficient") or are you asking why certain assumptions are needed to write the proof the textbook expects?
 
  • #3
Stephen Tashi said:
What assumptions would you propose to use instead?

Do you mean "necessary" in the mathematical sense (as in "necessary and sufficient") or are you asking why certain assumptions are needed to write the proof the textbook expects?

I guess what I'm struggling to understand is why the existence of the limit of f(g(x)) depends on these assumptions. For example, if limy→L f(y) did not exist or if g[I\{x}] was not contained in J\{g(x)} would that imply that the limit of f(g(x)) doesn't exist? If so, why?
 
  • #4
sonofagun said:
For example, if limy→L f(y) did not exist or if g[I\{x}] was not contained in J\{g(x)} would that imply that the limit of f(g(x)) doesn't exist?

No, it wouldn't.

For example, define [itex]f(x) [/itex] by [itex] f(x) = 0 [/itex] if [itex] x < 0 [/itex] and [itex] f(x) = 1 [/itex] if [itex] x \ge 0 [/itex].
[itex] Lim_{y \rightarrow 0} f(y) [/itex] does not exist.

Define [itex] g(x) = |x| [/itex]

[itex] lim_{y \rightarrow 0} f(g(y)) = 1 [/itex]

You haven't explained what you mean by "necessary".

There is a technical meaning for "necessary". The theorem that "A implies B" asserts that statement(s) A are "sufficient" for statement B to be true. To say that statement(s) A are "necessary" for B to be true asserts that "B implies A" .

There is a colloquial meaning for "necessary". By that meaning, we only assert that, having a certain method of proof in mind, the statements A are required to do that particular method of proof.

I don't know whether the "if..." part of the theorem you stated is "necessary" is the technical sense. The fact that theorem is stated in a text doesn't imply that the converse of the theorem is true.

( In the theorem, [itex] lim_{z \rightarrow x} g(x) = L \in J [/itex] should say "[itex] g(z) [/itex]" instead of [itex] g(x) [/itex] .)
 
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  • #5
Stephen Tashi said:
No, it wouldn't.

For example, define [itex]f(x) [/itex] by [itex] f(x) = 0 [/itex] if [itex] x < 0 [/itex] and [itex] f(x) = 1 [/itex] if [itex] x \ge 0 [/itex].
[itex] Lim_{y \rightarrow 0} f(y) [/itex] does not exist.

Define [itex] g(x) = |x| [/itex]

[itex] lim_{y \rightarrow 0} f(g(y)) = 1 [/itex]

You haven't explained what you mean by "necessary".

There is a technical meaning for "necessary". The theorem that "A implies B" asserts that statement(s) A are "sufficient" for statement B to be true. To say that statement(s) A are "necessary" for B to be true asserts that "B implies A" .

There is a colloquial meaning for "necessary". By that meaning, we only assert that, having a certain method of proof in mind, the statements A are required to do that particular method of proof.

I don't know whether the "if..." part of the theorem you stated is "necessary" is the technical sense. The fact that theorem is stated in a text doesn't imply that the converse of the theorem is true.

( In the theorem, [itex] lim_{z \rightarrow x} g(x) = L \in J [/itex] should say "[itex] g(z) [/itex]" instead of [itex] g(x) [/itex] .)

I was using the colloquial meaning of necessary.
 

FAQ: Why Are Assumptions Critical in the Limit of Composite Functions?

1. What is the definition of a limit of a composite function?

The limit of a composite function is the value that a function approaches as the input approaches a certain value. It is a mathematical concept used to describe the behavior of a function near a particular point.

2. How is the limit of a composite function calculated?

The limit of a composite function can be calculated by substituting the value of the input into the function and evaluating the resulting expression. If the resulting expression is indeterminate, further algebraic manipulation or the use of L'Hôpital's rule may be necessary to determine the limit.

3. Can the limit of a composite function exist even if the function is not defined at that point?

Yes, the limit of a composite function can exist even if the function is not defined at that point. This is because the limit is concerned with the behavior of the function near the point, not necessarily at the point itself.

4. How is the limit of a composite function used in real-world applications?

The limit of a composite function is used in a variety of real-world applications, such as physics, engineering, and economics. It is used to model and predict the behavior of systems and processes, and to optimize parameters for maximum efficiency.

5. Are there any special rules for finding the limit of a composite function?

Yes, there are a few special rules for finding the limit of a composite function. These include the sum and difference rules, product rule, quotient rule, and chain rule. These rules can make finding the limit of a composite function easier and more efficient.

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