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: Limit of integral

  1. May 23, 2010 #1
    1. The problem statement, all variables and given/known data

    are there any rules on how to find the limit of an integral equation?

    for example,

    find x such that the limit as y(x) tends to infinitiy of the integral equation equals 1
    [tex]lim \int_0^x \frac{1}{y(t)-y(x)}dt=1[/tex]

    2. Relevant equations

    3. The attempt at a solution

    Im not sure how to do this, can i simply swap the limit sign with the integral sign?

    Thanks in advance.
  2. jcsd
  3. May 23, 2010 #2


    User Avatar
    Homework Helper
    Gold Member



    For all values of [itex]x[/itex], so I'm not sure what you mean here.
  4. May 23, 2010 #3
    why does the limit = 0

    what if we had t in the numerator instead of 1?
  5. May 23, 2010 #4
    I think you're asking:


    [itex]\lim_{x\to a} y(x)=\infty[/itex]

    is there a function y(x) such that:

    [itex]\lim_{x\to a}\int_0^x \frac{1}{y(t)-y(x)}dt=1[/itex]
  6. May 23, 2010 #5
    no, i mean, i need to find an 'x'.

    I agree that the integrand is 0 (if y tends to infinity)

    but what if we have


    can we find an x such that the limit is 1?
  7. May 23, 2010 #6
    If you are fixing x, how can y(x) tend to infinity?
  8. May 23, 2010 #7
    because i need to find an x, say x* such that as y(x*) tends to infinity, the limit is 1
  9. May 23, 2010 #8
    y(x*) is some number. It doesn't make sense to say that a number tends to infinity. Perhaps I'm misunderstanding the question?
  10. May 23, 2010 #9
    sorry, i think im confusing things.

    forget y(x*).

    I want to find x such that

    so y(x) is just a function depending on a variable x. then, after i found the limit (in terms of x), i want to solve the equation for x...
    am i making sense?
  11. May 23, 2010 #10
    Would the following interpretation be correct:

    Look for x* such that

    \lim_{x \rightarrow x^*} y(x) = \infty
    \quad\text{ and }\quad
    \lim_{x \rightarrow x^*} \int_0^x \frac{(x-t)^{-1}dt}{y(t)-y(x)} = 1,

    where we always approach x* from the direction of 0.

    But even if this isn't the right interpretation, it seems important to know what y(x) is as well.
  12. May 23, 2010 #11
    yes, this is the right interpretation.
    is it possible to find x* without knowing what the function is?
  13. May 23, 2010 #12


    User Avatar
    Homework Helper
    Gold Member

    Are you sure your integrand isn't [tex]\frac{y(t)-y(x)}{t-x}[/tex] instead?
  14. May 23, 2010 #13
    yes, im sure.
    would it be easier if it was?
  15. May 23, 2010 #14
    In addition, I don't see how to find x* without knowing the function. I mean, x* appears at a vertical asymptote, but different functions have different asymptotes.
  16. May 23, 2010 #15
    lets assume it has a vertical asymptote.
    would we be able to find x* then?
  17. May 23, 2010 #16
    Evaluate the improper integral at most at two candidates for x*, one to the left and right of zero. Then, if one of those equals 1, you're good. I don't know another way.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook