Infinite Limits

Since it seems that the season of geekzoidness is in the air, Eternal Wanderer has decided that's safe to take a peep out of his stuffy nerd-closet after all. But he doesn't want it to be an ordinary, run-of-the-mill coming-out-of-the-nerd-closet rite. He wants it fabulous and bonggang-bongga ever!

So for his show and tell, he has decided to share with his readers why the limit of 1/x as x approaches 0 does not exist.

lim (1/x) = D.N.E.

x -->0

Ano daw???

*insert brain aneurysm here*

Wahahahahaha

Get your tissues and hankies out in case of epistaxis - here's a very informal proof:

Shall we look at this table?

x	1/x
2	1/2
1	1
.1	10
.01	100
.001	1000

etc.

Yes, you can probably guess that as x becomes smaller (approaching 0 from the right), the value of 1/x becomes bigger. To misquote Buzz Lightyear's words "to (positive) infinity and beyond!" However, x can't exactly be 0 because 1 divided by 0 is, ermmm, undefined.

Miss Minchin: I beg to differ. There such exist indeterminate forms that can be resolved through the application of l'Hôpital's rule...

Ternie: Stuff it, bitch. We'll save that for later!

I swear, Ms. Minchin is such a know-it-all nag-hag *rolls eyes*

But going back, for those of you who like looking at porn pictures, here's a graph of how it looks like when 0 is approached from the right side:

In very arcane abracadabra symbols, the whole idea can be written out like this:

lim (1/x) = +∞

x -->0+

(Martian translation alert #1: the limit of 1/x as x approaches 0 from the right is positive infinity.)

The same thing happens when you approach zero from the left side. As x moves towards zero (-2,-1,-.1, -.01, etc) without actually becoming 0, 1/x becomes more, ahmmm, "negative" (-1/2, -1, -10, -100, etc).

Miss Minchin: Let me point out something with your terminology.  In mathematical parlance, the comparative term "more negative" does not exist. May I suggest, perhaps, that you use the term...Ooooooh, wait. Was that a nubile young man I just saw passing by the classroom window?

Sigh. 

Yes, Miss Minchin can be a slutty stuff of a hag sometimes.

Anyhow, here's how it looks graphically:

See how the value of 1/x goes towards the negative infinity? Let's go back to abracadabra symbols and write 'em down like this:

lim (1/x) = -∞
x -->0-

(Martian translation alert #2: the limit of 1/x as x approaches 0 from the left is negative infinity.)

Interestingly, there's a gobbledygook, but extremely convenient theorem that says a "full" limit exists if and only if there's a limit on both the left and right sides. It also says that those "half" limits should be the same. But since we've already shown that the right-side limit is +∞, and that the left-side limit is -∞, we can safely say that the limit of 1/x as it approaches 0 does not exist.

Neat, ey?...But what's the whole point?

It's just that sometimes, it's we who put limits to ourselves. But do stop and take stock of the situation. If you take a closer look, the limit may not actually exist after all. Ease up. Let go. In taking down the concept of self-limitation, a whole new door of possibilities opens up before us.

O, san ka pa?!

I bow.

Chenkyu.