How can we define a limit approaching negative infinity?

  • #1
RubroCP
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I have the following definition:
$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$
From this, how can I get the definition of
$$\lim_{x\to p^-}=-\infty? $$
 
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  • #2
The definition says: We can make the values of ##f(x)## arbitrary great around ##x=p^+.## or: There is always a neighborhood of ##p^+## in which the function values become at least as high as we want them to be.

Now, how does this transform to low instead of high?
 
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  • #3
fresh_42 said:
The definition says: We can make the values of ##f(x)## arbitrary great around ##x=p^+.## or: There is always a neighborhood of ##p^+## in which the function values become at least as high as we want them to be.

Now, how does this transform to low instead of high?
Thanks, my friend. I will try here again.
 
  • #4
RubroCP said:
I have the following definition:
$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$
From this, how can I get the definition of
$$\lim_{x\to p^-}=-\infty? $$
The usual definitions don't use ##\epsilon## for this type of limit, but instead use M or some other letter. ##\epsilon## is typically some small number, relatively close to zero.

The definition you have usually goes like this:
$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall M >> 0, \exists \delta > 0, \text{ if } p \lt x \lt p + \delta \text{ then } f(x) > M$$

The definition for the limit being negative infinity is similar.
 
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  • #5
Mark44 said:
The definition you have usually goes like this:
limx→p+f(x)=+∞⟺∀M>>0,∃δ>0, if p<x<p+δ then f(x)>M

The definition for the limit being negative infinity is similar.
What about the "limit" of 1/x as x tends to 0? There is no interval around 0 where 1/x is >100.
 
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  • #6
Svein said:
What about the "limit" of 1/x as x tends to 0? There is no interval around 0 where 1/x is >100.
The definition I wrote is for a one-sided limit. For any x in the half-interval (0, .01), 1/x > 100.
 
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  • #7
OK. I would prefer using absolute values, though. And then there is the philosophical question: Is +∞ different from -∞? Since neither one represents a number, how can we tell?

Since my main background is complex analysis, I prefer the Riemann sphere mapping.
 
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  • #8
Svein said:
OK. I would prefer using absolute values, though. And then there is the philosophical question: Is +∞ different from -∞? Since neither one represents a number, how can we tell?
Saying that ##\lim_{x \to x_0} f(x) = \infty## is merely shorthand for saying that the function values get arbitrarily large as x gets nearer to ##x_0##. The function that you listed, f(x) = 1/x, can be made arbitrarily large and positive for positive x near zero, and can be made arbitrarily negative for negative x near zero. Neither limit "exists" in the sense of being a real number, but the infinity symbol conveys the unboundedness of the function values. The two one-sided limits are qualitatively and quantitatively about as different as they could possibly be, so, yes, ##+\infty## and ##-\infty## are very different.
 
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  • #11
Well, spring is here in force and my head is stuffed - some philosophical ideas should really be left undiscussed until pollen season is well and truly over.
 
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  • #12
The cardinality of an infinite set, and infinity or negative infinity being the value of a limit, are basically totally distinct concepts that happen to share the word infinity. I'm sure there's some theory out there that unites the concepts in a very cool way, but I think for a first pass at learning either concept it's best to totally ignore the other one.
 
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