1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Taylor's series limit 0/0

  1. Nov 23, 2011 #1
    1. The problem statement, all variables and given/known data

    I need to solve this limit in the form 0/0 with the Taylor series...

    2. Relevant equations

    3. The attempt at a solution
    Alright, I didn't really get where I am supposed to "stop" writing polinomials, my teacher said that I should stop when I find the smallest degree factor, because that's the one which is "bossing around" when the limit approaches zero.
    Okay, that's where I've gone so far:

    http://www4d.wolframalpha.com/Calculate/MSP/MSP619i577b6i0hb357g0000173a0a18c530af8g?MSPStoreType=image/gif&s=64&w=320&h=58 [Broken]

    I don't get if I wrote too many, if I didn't write enough terms, if I did something wrong at all, or I am right and should keep on doing calcs. Could someone help me out please? Thanks.
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Nov 23, 2011 #2

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Collect like terms in the numerator. What does that give you?

    And what about the denominator?
  4. Nov 23, 2011 #3
    [itex]-(2 x^4)/3-(119 x^6)/120-(1261 x^8)/5040[/itex]

    Even if I collect like terms, I get something that doesn't really get me close to the limit, and I think that huge number there is just wrong...
    But, I don't get when I have to stop, I mean, I could have gone for infinity keeping on writing the series polynomials. When do I have to stop writing?
  5. Nov 23, 2011 #4

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Remember L’ Hôpital’s Rule. If both f(x) and g(x) approach 0 at some point x0, then to evaluate f(x)/g(x) at x0 you try to evaluate f'(x)/g'(x) at x0. If that still results in the indeterminate form 0/0, you can iterate and try to evaluate f''(x)/g''(x). If that doesn't help, try to evaluate f'''(x)/g'''(x), and so on, until you either reach some form that is not indeterminate or a form that blows up.

    Assume that you have a Taylor expansion of f(x) and g(x) about the point of interest[tex]\begin{aligned}
    f(x) &= \sum_{n=0}^{\infty} a_n (x-x_0)^n \\
    g(x) &=\sum_{n=0}^{\infty} b_n (x-x_0)^n

    where the first few an and bn are zero. (If a0 and b0 are not zero there's no need for this L’ Hôpital rigamarole.)

    The limit is
    • Zero if the number of leading zeros in {an} is greater than the number of leading zeros in {bn}.
    • Undefined (infinite) if the number of leading zeros in {an} is less than the number of leading zeros in {bn}.

    These cases are kinda uninteresting. This leaves as an interesting case where the number of leading zeros in {an} and {bn} are equal.

    Which case applies to your problem?
  6. Nov 23, 2011 #5
    Okay, I tried to follow, the result of my limit is neither 0 nor infinity, so it must be when a_n or b_n are equal... uhm... how can I use such information to help myself into the problem?
  7. Nov 23, 2011 #6


    User Avatar
    Science Advisor
    Homework Helper

    The terms that are "bossing around" (i.e. are dominating as x->0) are the x^4 terms in the numerator and the denominator. Suppose you just look at those. What's the ratio?
  8. Nov 23, 2011 #7
    Ahh! Now I get it! :) I just checked with x^4 terms and it pops out -1/6 getting rid of the denominator x^4 term as well... thanks a lot! That's appreciated ! :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook