Calculus: Understanding Infinity in Functions

  • Context: MHB 
  • Thread starter Thread starter highmath
  • Start date Start date
  • Tags Tags
    Calculus Infinity
Click For Summary

Discussion Overview

The discussion revolves around understanding the concept of infinity in the context of functions, particularly in calculus. Participants explore how to determine the behavior of functions as they approach infinity, the meaning of limits, and the relationship between calculus and number theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how to ascertain that a function continues to increase or decrease as it approaches infinity.
  • Another participant clarifies that "infinity" is not a number and emphasizes the importance of discussing limits as x approaches infinity.
  • A claim is made that an increasing function does not necessarily have a limit of infinity as x approaches infinity, illustrated with the example f(x) = (x - 1)/x, which approaches 1.
  • A participant expresses confusion about how to determine the function's pattern as x approaches infinity and questions the utility of limits.
  • Another participant asks for clarification on what is meant by "function pattern."
  • A participant raises questions about the axioms needed to prove relationships in number theory and whether calculus can be used to demonstrate these relationships.
  • One participant corrects a misconception about number theory, stating it pertains specifically to positive integers and not to all real numbers, while also defining increasing and decreasing functions in terms of their derivatives.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of infinity in relation to functions, with some emphasizing the importance of limits while others question the definitions and implications. There is no consensus on the relationship between calculus and number theory, and the discussion remains unresolved regarding the axioms and proofs related to these concepts.

Contextual Notes

There are limitations in the definitions and assumptions regarding infinity and function behavior, particularly in how participants interpret the relationship between calculus and number theory. The discussion also reflects a lack of clarity on foundational concepts, which may affect the understanding of the topic.

highmath
Messages
35
Reaction score
0
When have a function and I know by investigation of that it getting "bigger and bigger" or getting "smaller and smaller", how could I know that in infinity it continue by that way always?
 
Physics news on Phys.org
highmath said:
When have a function and I know by investigation of that it getting "bigger and bigger" or getting "smaller and smaller", how could I know that in infinity it continue by that way always?
So the function is "increasing" or "decreasing". But I have no idea what "in infinity" means. In Calculus, "infinity" is not a number- it makes no sense to talk about the value of a function, or any property of a function "in infinity" or "at infinity". We can talk about the limit of a function "as x goes to infinity".

The most we can say here is that, if a function is increasing, then its limit as x goes to infinity is larger than or equal to any value of the function. If the function is decreasing then its limit as x goes to infinity is less than or equal to any value of the function.

If you are thinking that the limit, as x goes to infinity, of an increasing function must be infinity, that is incorrect. For example, if f(x)= (x- 1)/x= 1- 1/x then f(x) is increasing and the limit as x goes to infinity is 1.
 
Country Boy said:
function "as x goes to infinity".
.
If I know that x goes to infinity, so how can I know how the function pattern is there?
What the limit help me for?
 
First, you will have to tell us what you mean by "function pattern".
 
I know when you draw a function, the value of f(x) is real number always (generalization of natural (N), rational (Q), integer (Z) etc) on the Cartesian System.
So the question is the number theory.
o. k. I will continue with it.

(1)
What axioms I need to prove it?
By what can I use to show that the function is depend on Number Theory?
If I err tell me.
(2)
Is There a calculus way to prove it?
by what means in general?
 
I think you are confused about basic definitions. "Number theory" deals with specific properties of the positive integers. It is NOT about numbers in general and certainly not the set of all real numbers. In Calculus, a differentiable function is increasing at a given point if and only if its derivative is positive there and decreasing if and only if its derivative is negative.
 

Similar threads

Undergrad A question about limits and infinity
  • · Replies 7 ·
Replies
7
Views
2K
High School 1 divided by infinity equals zero (always?)
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
2K
Insights Series in Mathematics: From Zeno to Quantum Theory
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Looking for additional material about limits and distributions
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
2K
MHB Is infinity / infinity equal to 1?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad What would a calculus author have to say on ##\int r^2dm##?
  • · Replies 11 ·
Replies
11
Views
3K
High School Question about the fundamental theorem of calculus
  • · Replies 2 ·
Replies
2
Views
3K