How can we define a limit approaching negative infinity?

[RubroCP]

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
$$\lim_{x\to p^-}=-\infty? $$

 

[fresh_42]

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?

 

[RubroCP]

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.

 

[Mark44]

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.

 

[Svein]

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.

 

[Mark44]

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.

 

[Svein]

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.

 

[Mark44]

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.

 

[Svein]

The two one-sided limits are qualitatively and quantitatively about as different as they could possibly be, so, yes, +∞ and −∞ are very different.

https://en.wikipedia.org/wiki/Cardinality

 

[Mark44]

https://en.wikipedia.org/wiki/Cardinality

I understand the cardinality of infinite sets, but don't see that this is relevant when talking about limit values.

 

[Svein]

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.

 

[Office_Shredder]

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.