Just a question regarding limits

[truewt]

Can a limit of a function = infinity?

 

[mjsd]

I guess you can if you insist, but you will need to be very careful with how you define "infinity"... for there are many types of infinities. However, sensible definition that works in general may be difficult/if not impossible to find.

 

[Moo Of Doom]

In general, yes. The definition is as follows:

$$\lim_{x\to a}f\left(x\right) = +\infty \mbox{ if and only if for any } \epsilon > 0, \mbox{ there exists a } \delta > 0 \mbox{ such that if } 0 < \left|x - a\right| < \delta, \mbox{ then } f(x) > \epsilon .$$

EDIT:

And if you want the limit as $x$ approaches infinity, we simply modify the $\delta$ condition a little:

$$\lim_{x\to \infty}f\left(x\right) = +\infty \mbox{ if and only if for any } \epsilon > 0, \mbox{ there exists a } \delta > 0 \mbox{ such that if } x > \delta, \mbox{ then } f(x) > \epsilon .$$

EDIT2: Added 0 as a lower bound to make things correct... forgot about it, heh.

 

[truewt]

In general, yes. The definition is as follows:

$$\lim_{x\to a}f\left(x\right) = +\infty \mbox{ if and only if for any } \epsilon > 0, \mbox{ there exists a } \delta > 0 \mbox{ such that if } \left|x - a\right| < \delta, \mbox{ then } f(x) > \epsilon .$$

EDIT:

And if you want the limit as $x$ approaches infinity, we simply modify the $\delta$ condition a little:

$$\lim_{x\to \infty}f\left(x\right) = +\infty \mbox{ if and only if for any } \epsilon > 0, \mbox{ there exists a } \delta > 0 \mbox{ such that if } x > \delta, \mbox{ then } f(x) > \epsilon .$$

why is the condition necessary in defining the limits?

hmmm but i thought i learned that limits should not be =infinity, 'cause that's not defined as infinity isn't a real number..

 

[Moo Of Doom]

Infinity is an "extended real number." When talking about limits, it is often useful to be able to note in limit notation when a function blows up at a certain point, and in what direction.

It's necessary in addition to being sufficient simply so you can apply it to prove when a function does not have a limit, and also so you can apply this definition to prove things about functions that have limits.

 

[truewt]

so the conditions you posted only applies to limits =infinity?

but i can't seem to understand what those conditions mean..

mind explaining it in english?

 

[Moo Of Doom]

In English, we say that a function of $x$ approaches positive infinity as $x$ gets closer and closer to a point $a$ if for any number, no matter how large, you can find a positive distance from $a$ such that as long as $x$ is within that distance from $a$ the function is always larger than that number.

Another way to look at it is that if you look at a small interval around $a$, the minimum of the function on that interval (excluding $f\left(a\right)$) can be made as large as you want by decreasing the length of the interval.

Take for example, the function $f\left(x\right) = 1/x^2$ and look at it near zero. Is there an interval around zero, so that everywhere in the interval except 0, the function is larger than 1? How about larger than 10? 100? Is there a real number for which you cannot find such an interval?

 

[truewt]

Thanks a lot moo :)

 

[mathwonk]

this is merely a confusing matter of the use of language. most books say bluntly in the first section on limits that limits must be finite. then they contrdict themselves later by defining infinite limits.

so yes they can unless the author says he means not in a particular paragraph.

so yes it is possible, but no sometimes people restrict the use of the word limit to exclude reference to this case.

a nbhd of +infinity is an open interval of form (b, +infinity)

so a function has limit +infinity as x approaches a if for every b, there is a nbhd of a on which all values of f are larger than b.

 

[HallsofIvy]

I prefer to think that "$\lim_{x\leftarrow a} f(x)= \infty$" says that f(x) does not have a limit at a in a specific way.

 

[mathwonk]

what the heck does that mean halls? i gave a precise definition. or are you joking?

 

[ZioX]

An amusing thing with analysis is that saying the limit converges to infinity is precisely the same as saying the limit diverges.

 

[Gib Z]

I've never heard anyone say the limit converges to infinity in a formal sense though ..

I'm afraid I don't understand what Halls meant either >.< Although I'm sure Halls was just stating the way he likes to the of it when no rigor is required, I doubt he was contesting your definition mathwonk.

 

[cepheid]

I thought that a plain english way of interpreting:

$$\lim_{x\to a}f\left(x\right) = +\infty$$

was that the value of f(x) increases without bound as x approaches a. It doesn't actually approach any finite limit, (as ZioX hinted, the limit doesn't actually exist). Using notation that seems to say that the limit "exists" and "is equal to 'infinity'" is just a shorthand way of saying, "the function blows up at a."

Am I right?

(Moo of Doom's precise def'n seems to indicate that I am right...at least the way I am reading it).

EDIT: I think Mathwonk's does too, but I have no idea what nbhd is.