Wednesday, April 21, 2010

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*


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

Shall we look at this table?














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?


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.



  1. and that my dear, is what you call mono no aware.


  2. um... round to the nearest 10, group by 100s and work in pencil.

  3. that's why there are asymptotes. ano daw?

  4. This comment has been removed by the author.

  5. Engel: isn't calculus lurve? lolz teehee

    Mugen: i know you hate math, but ive it a try. fun sya noh! ;)

    PKF: go go go!

    John Stan: ay. nerd ka rin pala wahahahaha

    Red: why did you delete you comment?!

  6. oh my god, i think i just peed a little. ang sakit ng ulo ko trying to process yung equation. buti nalang naintindihan ko yung practical application sa dulo.

  7. My eyes haven't glazed over that badly since the last time I attended Sunday Mass.

    Back when Cory was still President.

  8. oh my god what is this? im so bobo naman here

  9. Yes, however apart from the L'Hôpital's rule and/or algebraic elimination, other methods can be used to manipulate the expression so that the limit can be evaluated. :-)

  10. I felt my comment was too rude, too brash and disrespectful. This is your home, and its your turn to release your inner geekness, not me.

    Plus, the argument was simply contestable. Even the concept of infinity was devised in convenience to designate numerically problematic expressions. Otherwise, I prefer not work with the extended set of real numbers. They're just too shifty and vague.

  11. Coño: i'll tutor you in math and OTHER stuff ;)

    Iuri: isa ka pang nerd wahahahaha

    Red: oh, no offense taken at all :D

    p.s. sure, in extended real numbers, the lim 1/x as x -> 0 = ∞, but let's not go into that lolz

  12. Will not even dare add to the erudition of the others... Bow na lang po! Bow!

  13. i have no love for math. pero i had 20 plus units of it in college.