Limiting x->0: Analyzing f(x) = 1/|[[x]]|

  • Thread starter Thread starter madah12
  • Start date Start date
Click For Summary

Homework Help Overview

The discussion centers around evaluating the limit of the function f(x) = 1/|[[x]]| as x approaches 0, where [[x]] denotes the largest integer function. Participants are exploring the behavior of the function near the point of interest and the implications of the greatest integer function and absolute values.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are attempting to redefine the absolute value of the greatest integer function and are questioning the behavior of the function as x approaches 0 from both the left and the right. There is confusion regarding the values of [[x]] for small positive and negative inputs.

Discussion Status

The discussion is ongoing, with participants providing insights and clarifications about the greatest integer function and its implications for the limit. Some guidance has been offered regarding evaluating the limits from both sides, but no consensus has been reached on the values of the limits.

Contextual Notes

Participants are grappling with the definitions and behaviors of the greatest integer function and absolute values, particularly in the context of approaching zero. There is a noted ambiguity in interpreting the limits based on the properties of the function.

madah12
Messages
326
Reaction score
1

Homework Statement


Limit x->0 f(x)
where f(x) = 1/|[[x]]|
where [[x]] is the largest integer function

Homework Equations





The Attempt at a Solution


I am having problem redefining the absolute value here
|[[x]]| = -[[x]] when [[x]] <0
so that is when x<0?
the right hand limit is supposed to be infinity
but I am confused about the left one is it 1?
or is it negative infinity or infinity?
 
Physics news on Phys.org
You are overthinking this. To take the "limit as x goes to 0", look at points close to x= 0. What is [[0.00001]]? What is [[-0.000001]]? What are the absolute values of those?

Do "limit from the right" (x> 0) and "limit from the left" (x< 0) separately.
 
it isn't the absolute value function its the greatest integer function
http://www.icoachmath.com/SiteMap/GreatestIntegerFunction.html
but my problem is that since .9999999 continuously is 1 should I consider the greatest integer of -.00000000000...01 as 0 or -1?
 
Last edited by a moderator:
You said it was "the absolute value of the greatest integer function".

And the greatest integer less than or equal to 0.0000000...01 (where the ... must mean some specific number of missing 0s, not an infinite string) is 0, of course. Since it is positive why would you even consider -1? That has nothing to do with "0.9999999..." being equal to 1.
 
the limit from the right is 0 and the left its 1 right?
 
Last edited:

Similar threads

##\lim_{x \to0} \left(\dfrac{1}{\sin^2 x}-\dfrac{1}{x^2}\right)##
  • · Replies 5 ·
Replies
5
Views
2K
Use Graph To Investigate Limit
  • · Replies 10 ·
Replies
10
Views
1K
Is the given function continuous at x= 0? f(x) = {sin(1/x) if x≠0, 1
  • · Replies 4 ·
Replies
4
Views
4K
Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
2K
Proving piecewise function is k-differentiable
  • · Replies 5 ·
Replies
5
Views
2K
Calculate the limit cos(x)/sin(x) when x approaches 0
  • · Replies 12 ·
Replies
12
Views
4K
Differentiable function proof given ##f''(c) = 1##
  • · Replies 1 ·
Replies
1
Views
1K
Prove that Lim x->c f(x)g(x)=0 if Lim x->c f(x)=0 and |g(x)|<K
  • · Replies 2 ·
Replies
2
Views
2K
Prove that the limit of |x|/x at x=0 DNE
  • · Replies 9 ·
Replies
9
Views
3K
Use the graph of f(x) to investigate the limit
  • · Replies 6 ·
Replies
6
Views
2K