Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

  • Thread starter Thread starter zenterix
  • Start date Start date
  • Tags Tags
    Calculus Limits
Click For Summary

Homework Help Overview

The discussion revolves around a problem from Spivak's Calculus concerning the behavior of a function defined by modifying decimal expansions, specifically focusing on limits and graph behavior. The function replaces digits in the decimal expansion of a number after the first occurrence of the digit 7 with zeros. Participants are exploring the implications of this function on limits and the nature of the graph, particularly around points with decimal expansions ending in 7's.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to understand the behavior of the function at specific points, particularly those with decimal expansions that include or exclude the digit 7. Questions arise regarding the existence of limits at certain values, such as 0.73 and 0.8, and the implications of the function's definition on these limits. There is also discussion about the nature of intervals on the graph and the reasoning behind open versus closed intervals.

Discussion Status

The discussion is ongoing, with participants sharing their interpretations and questioning the assumptions underlying the function's behavior. Some have provided initial proofs and reasoning for the existence of limits at certain points, while others are seeking clarification on specific aspects of the function and its graphical representation. There is a productive exchange of ideas, but no consensus has been reached on all points raised.

Contextual Notes

Participants are navigating the conventions of decimal representation and the implications of these conventions on the function's behavior. There are references to the solution manual's graph, which includes open intervals, prompting questions about the correctness of these representations. The discussion also touches on the definitions and properties of limits in the context of this specific function.

  • #31
PeroK said:
The main proof as you call it is potentially important, as it formalises the argument. But, without the side proof it means nothing.

That proposition must be false. The problem is that a small reduction in ##x## after a string of zeroes may cause a long string of nines. Using the double digit idea solves this problem.
The proposition should actually be with a ##\leq##, ie ##a-\delta \leq f(x)##. Is this what you mean?
 
Physics news on Phys.org
  • #32
zenterix said:
The proposition should actually be with a ##\leq##, ie ##a-\delta \leq f(x)##. Is this what you mean?
You're right. But, I don't see how that proposition helps.
 
  • #33
PeroK said:
This is effectively what you need to prove. Otherwise your ϵ−δ is just window-dressing!
It's the proof of this, which was a step in the main proof. It's required for the latter.
 

Similar threads

Spivak, Ch 5 Limits, Problems 10c: Proving limit relationship
  • · Replies 5 ·
Replies
5
Views
2K
Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds
  • · Replies 6 ·
Replies
6
Views
2K
Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function
  • · Replies 16 ·
Replies
16
Views
2K
[Spivak Calculus, Ch. 5 P. 9] Showing equality of two limits
  • · Replies 2 ·
Replies
2
Views
2K
Understanding the solution to a calculus problem (removable discontinuities)
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
3K
Graphing Derivatives: Concavity and Inflection Points
  • · Replies 4 ·
Replies
4
Views
3K
Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1
  • · Replies 8 ·
Replies
8
Views
2K
Prove that function tends to 0 everywhere in this interval
  • · Replies 4 ·
Replies
4
Views
2K
MHB Can you help with these calculus questions?
  • · Replies 4 ·
Replies
4
Views
2K