The discussion revolves around finding the limit of the function \(\frac{x^2}{1+x^2}\) as \(x\) approaches negative infinity. Participants explore the implications of the limit definition and the conditions under which it holds, focusing on the mathematical reasoning involved in establishing the limit.
Participants express differing views on the approach to finding \(N\) and the reasoning behind selecting roots, indicating that multiple perspectives and methods are being considered without a consensus on the best approach.
Some participants note that the approach to finding \(N\) could be unconventional and that the reasoning for selecting the negative root may require further exploration. There are also unresolved steps in the mathematical reasoning presented.
Finding "a negative number N such that if x< N then ..." is very peculiar. But since x only occurs to even power, it doesn't matter whether you use x or -x. "Find a positive number M such that if x> N then |\frac{-1}{1+x^2}|< epsilon" is more "standard" and gives N= -M. Saying that |\frac{-1}{1+x^2}|< \epsilon is the same as saying 0< \frac{1}{1+x^2}< \epsilon which is, in turn, the same as saying that 1+ x^2> \frac{1}{\epsilon} which is the same as x^2> \frac{1}{\epsilon}- 1. Can you carry on from there?John O' Meara said:Find a negative number N such that |\frac{x^2}{1+x^2}-1| < \epsilon for x < N. That implies |\frac{-1}{1+x^2}| < \epsilon if x < N.This gives me the following -|\frac{1}{1+x^2}| < \epsilon if x < N.I do not know what to do from here. Please help as I am teaching myself. Thanks.