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

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In summary: U of a there exists a neighborhood V of b such that f(g-1(V\{b}))⊆U. This is very similar to the limit as x→+∞ and as x→-∞, which are both stated as f(g-1(V\{b}))⊆U. It seems that the notation is generalized in the following way: if for any neighborhood U of a there exists a neighborhood V of b such that f(g-1(V\{b}))⊆U, then the notation "limit as |x| goes to infinity" is also valid.Is there any difference when evaluating them?
  • #1
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
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
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
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
Cristopher said:
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’
 

1. What is the difference between lim as x→∞ and lim as |x|→∞?

The main difference is that in the first case, the limit is approaching infinity from the positive side, while in the second case, the limit is approaching infinity from both positive and negative sides. In other words, the absolute value ensures that the limit is taken from both directions, while in the first case, only the positive direction is considered.

2. How do the graphs of lim as x→∞ and lim as |x|→∞ differ?

The graph of lim as x→∞ is a horizontal line at y = L, where L is the limit, while the graph of lim as |x|→∞ is a horizontal line at y = L, but with a "hole" in the middle at x = 0. This is because the limit is undefined at x = 0 since the function approaches different values from both sides.

3. Can lim as x→∞ and lim as |x|→∞ have different values?

Yes, it is possible for these two limits to have different values. This is because in the first case, the limit is only considering the positive direction, while in the second case, the limit is considering both positive and negative directions. If the function approaches different values from these directions, then the limits will also be different.

4. How can we determine the limit as |x|→∞ algebraically?

To determine the limit as |x|→∞ algebraically, we can write out the limit expression and then use the properties of limits to simplify it. For example, if we have lim as |x|→∞ of (2x + 1), we can use the property of limits that states that if the limit of f(x) exists, then the limit of kf(x) also exists and is equal to k times the limit of f(x). In this case, the limit would be equal to 2 times the limit as x→∞ of x, which is equal to infinity.

5. What is the significance of distinguishing between lim as x→∞ and lim as |x|→∞?

Distinguishing between these two limits is important because it allows us to understand how the function behaves in different directions as it approaches infinity. In some cases, the limit as x→∞ may exist and be finite, but the limit as |x|→∞ may not exist or be infinite. This can give us insight into the behavior of the function and help us make accurate conclusions about its properties.

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