Find lim (x^2n - 1)/(x^2n + 1) x->infinity->

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Discussion Overview

The discussion revolves around evaluating the limit of the expression (x^2n - 1)/(x^2n + 1) as x approaches infinity. Participants explore various approaches and implications of the limit, including considerations for different values of n and the behavior of the function under different conditions.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants suggest substituting x=1/h to analyze the limit as h approaches zero, leading to a proposed limit of 1.
  • Another participant notes that the expression can be simplified to (B - 1)/(B + 1) for large B, indicating that the terms -1 and +1 become negligible.
  • One participant presents a guess about the limit's value depending on the range of x, suggesting it equals 1 for x > 1, -1/2 for x = 1, -1 for 0 <= x < 1, and undefined for x < 0.
  • Another participant raises the question of what happens when n approaches infinity, prompting further discussion.
  • There is a correction regarding the original question, emphasizing that it asks for the limit as x approaches infinity, not n.
  • A later reply attempts to clarify the limit for various cases, suggesting that it equals 1 if |x| > 1, 0 if |x| = 1, and -1 if |x| < 1, while asserting that it is not undefined for any values.
  • Participants acknowledge confusion about the context of n approaching infinity versus x approaching infinity, with some expressing a lack of clarity on the original question.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the limit, particularly concerning the implications of varying n and the conditions under which the limit is evaluated. The discussion remains unresolved with no consensus on the limit's value as x approaches infinity.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the values of n and x, as well as the conditions under which the limit is evaluated. Some participants' responses depend on specific interpretations of the problem, leading to different conclusions.

lizzie
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find lim (x^2n - 1)/(x^2n + 1)
x->infinity
-> means tends to
 
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lizzie said:
find lim (x^2n - 1)/(x^2n + 1)
x->infinity
-> means tends to

Have you tried substituting x=1/h so as x->infinity, h->zero.
Using this I got the answer as 1
 


This is just (B - 1)/(B + 1) for some B that gets really big (goes to infinity with x). For all problems of this form you can just divide out the B and you'll see that the little stuff like +1 and -1 drop out.
 


lizzie said:
find lim (x^2n - 1)/(x^2n + 1)
x->infinity
-> means tends to

Off the top of my head, my best guess is ...

For real numbers, the limit is equal to 1 if x > 1, -1/2 if x = 1, -1 if 0<=x<1, and is undefined if x<0.

For complex numbers with non-zero imaginary part, the limit is equal to -1 if |x| < 1 and is undefined if |x| >= 1.
 


what will happen if n-> infinity
 


You asked that originally and you have already been given 4 answers.
 


DeaconJohn said:
Off the top of my head, my best guess is ...

For real numbers, the limit is equal to 1 if x > 1, -1/2 if x = 1, -1 if 0<=x<1, and is undefined if x<0.

For complex numbers with non-zero imaginary part, the limit is equal to -1 if |x| < 1 and is undefined if |x| >= 1.

HallsofIvy said:
You asked that originally and you have already been given 4 answers.

The original question asked for x--> infinity, not n.

As for the original question- Try adding and then subtracting 2 off the numerator.

For when n --> infinity, DJ had an attempt but needs some corrections: the limit is equal to one if |x| > 1, 0 if |x|=1, -1 if |x| < 1 and not undefined for any values. Note that we have an even function, so none of the "undefined if x<0" stuff.
 
Last edited:


Gib Z said:
The original question asked for x--> infinity, not n.

As for the original question- Try adding and then subtracting 2 off the numerator.

For when n --> infinity, DJ had an attempt but needs some corrections: the limit is equal to one if |x| > 1, 0 if |x|=1, -1 if |x| < 1 and not undefined for any values. Note that we have an even function, so none of the "undefined if x<0" stuff.

Gib Z, You are absolutely correct. My answer is in the context of n --> infinity. Funny thing is that I thought I was addressing the question of when x --> infinity. A credit to your understanding to realize that my calculations assumed n --> infinity. Unfortunately, I don't have anything to say about the case when x --> infinity, and the time that I've alloted to spend on this interesting problem of yours has expired.

I think I'll probably be concentrating on the number theory board in the future. That is the area of math that interests me most these days. It is also the area I know least about.

DJ
 

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