Difference between lim as x→∞ and lim as |x|→∞

  • Thread starter Cristopher
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I came across something I'd never seen before, the use of |x| instead of just ‘x’ in limits.
What is the difference between [itex]\displaystyle\lim_{|x|\to\infty}x\sin\frac{1}{x}[/itex] and [itex]\displaystyle\lim_{x\to\infty}x\sin\frac{1}{x}[/itex] ?
Is there any difference when evaluating them? Is that notation used only with infinity?

Thanks.
 

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  • #2
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It seems like it's implying that the limit as x goes to infinity is equal to the limit as x goes to negative infinity. But someone who has actually seen this notation before might know better, I'm just guessing.
 
  • #3
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I haven't seen limits as |x|→∞, but I have seen limits as x→±∞ or limits as x→∞ with this meaning. In the latter case they distinguished limits as x→+∞ and as x→-∞. I think they pictured ∞ as a single point outside the line, making it a circle, its Alexandroff compactification, considering limits as x→+∞ and as x→-∞ as the lateral limits towards ∞.

About the limits as |x|→∞, I think it could be generalized in the following way. We may say that f(x)→a as g(x)→b if, for any neighborhood U of a, there exists a neighborhood V of b such that f(g-1(V\{b}))⊆U. Also f(x)→a as g(x)→∞ if, for any neighborhood U of a, there exists M>0 such that f(g-1((M,∞)))⊆U, and similar definitions.
 
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  • #4
HallsofIvy
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I see two different ways of interpreting "limit as |x| goes to infinity". First would be that "limit as x goes to infinity" and"limit as x goes to -infinity" must be the same. The other would be that x represents a point in the plane or a complex number and x goes away from the origin in any direction.
 
  • #5
What is the difference between [itex]\displaystyle\lim_{|x|\to\infty}x\sin\frac{1}{x}[/itex] and [itex]\displaystyle\lim_{x\to\infty}x\sin\frac{1}{x}[/itex] ?
Is there any difference when evaluating them? Is that notation used only with infinity?
The first notation only really makes sense if you consider 'x' to be a complex variable. In this case, taking the limit at infinity means asking whether the function approaches a fixed value no matter which direction you approach infinity from.

The second notion on the other hand can be used if x is real and you are simply going to "positive" infinity.

At infinity, functions of complex variables generally either have a finite limit, or else a pole. They could also have an essential singularity (O~o) as well. And if the function is multi-valued, it will almost always have a (irregular) branch point there.
 
  • #6
Thank you all.

Yes, I also thought that it says that the limit as x→+∞ and as x→-∞ are the same. Indeed, I did look at the graph of the funcion x sin(1/x) before asking, and these limits are both equal to 1. I also noticed there's symmetry in the graph, I thought it may have something to do with that, too. I posted the question in the hope of getting more info on the scope of this notation.

Just for reference I saw the notation here:
http://en.wikipedia.org/wiki/L%27Hopital%27s_rule
In the section ‘Other ways of evaluating limits’
 

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