# Infinite limit definition

• I
• RubroCP
In summary, the conversation discusses the definition of a limit where the function approaches positive or negative infinity. The definition states that for the limit to be positive infinity, the function values can be made arbitrarily large around the limit point, and for the limit to be negative infinity, the function values can be made arbitrarily small around the limit point. The conversation also briefly touches on the concept of cardinality and its relevance to limit values.f

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

Delta2
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.

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.

RubroCP and Delta2
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.

RubroCP
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.

RubroCP and Delta2
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.

RubroCP
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.

RubroCP and Delta2
RubroCP
I understand the cardinality of infinite sets, but don't see that this is relevant when talking about limit values.

RubroCP and pbuk
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.

RubroCP
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.

RubroCP and pbuk