Confused About Infinity: Help Understanding a Math Limit

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Discussion Overview

The discussion revolves around understanding the mathematical limit as \( n \) approaches infinity, specifically the expression \(\lim_{n\rightarrow\infty} \frac{n(n+1)}{2n^2}\). Participants are exploring the implications of infinity in this context and how it affects the limit calculation.

Discussion Character

  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the limit and its relation to infinity.
  • Another participant provides a breakdown of the limit calculation, showing the steps leading to \(\frac{1}{2} + \frac{1}{2n}\).
  • A different participant suggests that as \( n \) approaches infinity, \( \frac{1}{n} \) approaches 0, and discusses the implications of multiplying by constants in limits.
  • There is a reiteration of the idea that \( 2 \times \text{infinity} = \text{infinity} \), indicating a potential misunderstanding of how infinity behaves in limits.

Areas of Agreement / Disagreement

The discussion shows some participants attempting to clarify the limit, but there is no consensus on the understanding of infinity and its implications in this context. Multiple interpretations and levels of understanding are present.

Contextual Notes

Some participants may be missing foundational assumptions about limits and infinity, leading to confusion in their reasoning. The discussion does not resolve these misunderstandings.

danne89
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My book tolds me that: [tex]\lim_{n\rightarrow\infty} \frac{n(n+1)}{2n^2}= \frac{1}{2}[/tex]. I don't get it. Maybe this with infinity, I dunno... Please, help!
 
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Because

[tex]\frac{n(n+1)}{2n^2} = \frac{n^2+n}{2n^2} = \frac{n^2}{2n^2} +\frac{n}{2n^2} = \frac{1}{2} + \frac{1}{2n}[/tex]
 
Ok, I think I got it now. So it's based on that 1/n equals 0, when n approaches infinity and 2 * infinity = infinity, right?
 
danne89 said:
Ok, I think I got it now. So it's based on that 1/n equals 0, when n approaches infinity and 2 * infinity = infinity, right?

yeah, the 1/(2n)=(1/2)(1/n) so you can factor the constant out of the limit, ie lim (1/(2n)) = (1/2) * lim (1/n) --> 0 as n --> infinity
 

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