Limits of Infinity: Does f(x) Exist?

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    Infinity Limits
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Discussion Overview

The discussion revolves around the conditions under which the limit of a function as x approaches infinity exists. Participants explore whether both the limits as x approaches positive and negative infinity must exist and be equal for the overall limit to be defined.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that for the limit as x approaches infinity to exist, both the limits as x approaches positive and negative infinity must exist and be equal.
  • Others argue against this, citing the example of f(x) = Arctan(x) as a counterexample.
  • A participant questions the reasoning by referencing a calculus text, suggesting that a specific limit does not exist for similar reasons.
  • Another participant clarifies that the limit as x approaches infinity is specifically the limit as x approaches positive infinity.
  • There is a discussion about conventions regarding infinity, noting that in standard real number analysis, infinity typically refers to positive infinity, while mentioning the concept of one-point compactification.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views regarding the conditions for the existence of limits at infinity remain unresolved.

Contextual Notes

There are limitations regarding the assumptions made about the definitions of limits and the conventions used in different mathematical contexts, which have not been fully explored in the discussion.

phymatter
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Is it so that for limx->infinity f(x) to exist ,
limx->+infinity f(x) and limx->-infinity should exist and be equal ? if so then why ?
 
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phymatter said:
Is it so that for limx->infinity f(x) to exist ,
limx->+infinity f(x) and limx->-infinity should exist and be equal ? if so then why ?

No, it isn't so. Look at f(x) = Arctan(x).
 
LCKurtz said:
No, it isn't so. Look at f(x) = Arctan(x).

Are you sure , because i.a. maron single variable calculus says that limx->infinity (2x2+3)1/2/(4x+2) does not exist for the same reason .
 
[tex]\lim_{x\to \infty}[/tex] IS the limit as x goes to positive infinity.
 
phymatter said:
Are you sure , because i.a. maron single variable calculus says that limx->infinity (2x2+3)1/2/(4x+2) does not exist for the same reason .

[tex]\lim_{x \to \infty}\frac{\sqrt{2x^2 + 3}}{4x + 2} = \frac{1}{4}[/tex]

Is I.A. Maron the author of a calculus text? I never heard of it.
 
You have to be careful with your conventions. Normally, with the real numbers, "[itex]\infty[/itex]" means "[itex]+\infty[/itex]. However, with the "one-point compactification" of the real numbers (not commonly used for Calculus problems!) we have just the one "point at infinity".
 

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