Understanding Limits: A Question about Inequalities

  • Context: Undergrad 
  • Thread starter Thread starter chemistry1
  • Start date Start date
  • Tags Tags
    Limit
Click For Summary

Discussion Overview

The discussion revolves around the concept of limits in calculus, specifically focusing on the inequality relationship between two functions, f(x) and g(x), as they approach a point c within an interval (a, b). Participants explore the implications of the inequality f(x) ≤ g(x) and how it relates to the limits of these functions as x approaches c.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the limit inequality, questioning whether the inequality f(x) ≤ g(x) holds at the point c, despite it being true for all x in the interval (a, b) except possibly at c.
  • Another participant clarifies that the phrase "except possibly at c" indicates that the limit inequality will generally hold, but the behavior at c may vary depending on the specific values of f(c) and g(c).
  • A different participant raises a question about the consideration of multiple functions on the same graph and whether the combination of these functions needs to be a single function.
  • There is a brief exchange where one participant asks for clarification on the previous question about combining functions, but the original poster indicates that they no longer need that clarification.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the implications of the limit inequality at the point c, and there is no agreement on the necessity of the combined functions being a single function.

Contextual Notes

The discussion includes assumptions about the behavior of functions at specific points and the interpretation of limit definitions, which may not be fully resolved.

chemistry1
Messages
108
Reaction score
0
Hi, I'm having trouble understanding the following fact about limits :
If f(x)<=g(x) for all x on (a,b) (except possibly at c) and a<c<b then,
lim f(x) <= lim g(x)
x -> c x->c
Here's how I interpret the definition : We have two functions f(x) and g(x), and the inequality f(x)<=g(x) hold true for all values that are not c. (That our interval (a,b)) If we were to evaluate the functions at c (considering that we can do it for our two functions.) then the inequality wouldn't hold anymore. (For example, f(x) would be superiro to g(x))
Please tell me if I have any errors.
THank you!
If you want to read more, go here : http://tutorial.math.lamar.edu/Classes/CalcI/ComputingLimits.aspx
 
Physics news on Phys.org
The definition includes the phrase "except possibly at c". This means the limit inequalty will hold. At c the inequalty may or may not hold delepnding on the definition.

Example: f(x) = 1 for x ≠ c, f(c) = k. g(x) = 2 for all x. Then the limits as x -> c satisfy f(x) < g(x). However at c it will depend on whether or not k > 2.
 
I was wondering, when we consider several functions at once in the same graph, is it ok if this whole is not a function itself ? Do we care about whether this whole is function or not ?
 
What "whole" are you talking about? How are you combining these "several functions"?
 
Nah, its okay, no need for that anymore.
 

Similar threads

Undergrad Some questions on l'Hôpital rules
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Proving that convexity implies second order derivative being positive
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Limit of the product of these two functions
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad What is the difference between these two power series theorems?
  • · Replies 5 ·
Replies
5
Views
3K
High School Proof of Chain Rule: Understanding the Limits
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Uniform convergence and derivatives -- difference between two theorems?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Substitution in a definite integral
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Definition of Limit for vector fields
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad What is the proof for nested limits?
  • · Replies 5 ·
Replies
5
Views
3K