Need help with a limit problem on Spivak's Calculus.

In summary, the problem states that for a function f(x) defined as 1/n if x is in A_n and 0 if x is not in A_n for any n, the limit of f(x) as x approaches a is equal to 0 for all a in [0,1]. The conversation discusses different approaches to proving this limit, including the possibility of finding a neighborhood of a that does not contain any elements of the A_n sets. However, it is noted that this may not always be possible due to the infinite number of sets involved. Ultimately, it is suggested to modify the original approach using cases.
  • #1
Elvz2593
5
0
1. Homework Statement

Suppose that [itex]A_{n} [/itex] is, for each natural number n, some finite set of numbers in [0,1], and that [itex] A_{n} [/itex] and [itex] A_{m} [/itex] have no members in common if m =/= n. Define f as follows:
[itex]
f(x)=\left\{\begin{array}{cc}1/n,&\mbox{ if }
x \in A{n} \\0, & \mbox{ if }x Not In A_{n} For Any n.\end{array}\right.
[/itex]

Prove that [itex]\lim[/itex] x-a f(x)=0 for all a in [0,1].

Michael Spivak - Calculus Ch5 Q24

Homework Equations



The Attempt at a Solution


I've been trying to figure out the existence of this limit, but so far with no luck. I might have misunderstood the problem. Can someone help me out with this?Update: Just looked a older post on this problem. I think I overlooked some crucial stuff.
So, if we pick a delta value so small that none of the points in A1,A2...An are on the (a-delta, a+delta) interval, except maybe point a itself. f(x) approaches 0 to as x -> a, even if f(a) is defined at some 1/n. Am I being correct?
 
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  • #2
Not quite. Remember, although there are only a finite number of elements in each [itex] A_n [/itex], there are an infinite number of the sets, one for each positive integer. So there's no "end" to the list of sets: [itex] A_1, A_2, \ldots [/itex]. So I claim you may not be able to always find such an interval. For example, let [itex] A_n = \{1/n\}[/itex] for each [itex] n \in \mathbb{Z}^+[/itex]. I claim that any neighborhood of 0 contains a point in [itex] A_n [/itex] for some n.

I think it might help to use the sequential version of functional limits.

EDIT: Oh, and welcome to PF!
 
  • #3
I think it is probably the author's wording error, otherwise it would be proved false.

Thank you so much for helping me out there. I like this place already.:biggrin:
 
  • #4
Elvz2593 said:
I think it is probably the author's wording error, otherwise it would be proved false.

No no, that wasn't a counterexample! As you'll note, we also have [itex] \lim_{n \to \infty} 1/n = 0 [/itex], so the proposition holds in this case as well. (Spivak is a very good book, and I'm sure he wouldn't make such an error!) I just wanted to show that you can't always find an interval that doesn't hit a point in any of the An's.

Thank you so much for helping me out there. I like this place already.:biggrin:

Glad to hear it!
 
  • #5
I see what you are saying now. So, the proposition would still hold as 1/n is approaching 0 as well.
 
  • #6
After thinking about it some more, I think we can modify your original approach using cases. In the first case we can find a neighborhood of the point that doesn't contain any elements of the [itex] A_n 's[/itex] and we're done. Otherwise, what does that say about each neighborhood of a? What must each one contain?
 

1. What is a limit in calculus?

A limit in calculus is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It represents the value that the function is "approaching" as the input gets closer and closer to the specified value.

2. How do you solve a limit problem in calculus?

To solve a limit problem in calculus, you can use various techniques such as substitution, algebraic manipulation, and the use of limit laws. It is also important to understand the properties and definitions of limits to correctly evaluate them.

3. What is Spivak's Calculus?

Spivak's Calculus is a well-known textbook written by Michael Spivak that covers the fundamental concepts of calculus, including limits, derivatives, and integrals. It is often used as a reference or textbook in college-level calculus courses.

4. What are some common challenges when solving limit problems?

Some common challenges when solving limit problems include identifying the type of limit (finite, infinite, one-sided), applying limit laws correctly, and recognizing indeterminate forms. It is also important to check for continuity and to consider the behavior of the function near the specified value.

5. Can you provide an example of solving a limit problem from Spivak's Calculus?

Yes, for example, the limit problem could be:
limx→3 (x²-3x+2)
To solve this, we can factor the expression to get:
limx→3 (x-1)(x-2)
Then, we can plug in the value 3 for x to get:
(3-1)(3-2) = 2
Therefore, the limit is 2.

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