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Limit problem

  1. Nov 13, 2015 #1
    1. The problem statement, all variables and given/known data

    Let ##f## be piecewise continuous from ##[0,+\infty[## into ##V = \mathbb{R} ## or ##\mathbb{C}##, such that ## f(x) \longrightarrow_{ x\rightarrow +\infty} \ell ##.

    Show that ## \frac{1}{x}\ \int_0^x f(t) \ dt \longrightarrow_{ x\rightarrow +\infty} \ell##

    2. Relevant equations

    Integration of asymptotic comparisons

    3. The attempt at a solution

    Can you tell me if this is correct please ?

    Since ##f - \ell = o_{+\infty}(1) ##, and since ## u: x \rightarrow 1 ## is non-negative, piecewise continuous, and non-integrable on ##[0,+\infty[##, then

    ## \int_0^x f(t) - \ell \ dt = o_{+\infty}(\int_0^x u(t) \ dt)##

    which is the same as saying that ##\int_0^x f(t) \ dt - x \ell = o_{+\infty}(x) ##.

    Multiplying left and right by ##\frac{1}{x}##, I get that ## \frac{1}{x}\ \int_0^x f(t) \ dt - \ell = o_{+\infty}(1)## which proves that

    ## \frac{1}{x}\ \int_0^x f(t) \ dt \longrightarrow_{ x\rightarrow +\infty} \ell##.

    Is this OK ?
     
  2. jcsd
  3. Nov 13, 2015 #2
    Never mind, I've had confirmation. Thanks !
     
  4. Nov 13, 2015 #3

    Mark44

    Staff: Mentor

    I think this is nicer notation: ##\lim_{x \to \infty} f(x) = \ell##
    My LaTeX script is ##\lim_{x \to \infty} f(x) = \ell##
     
  5. Nov 13, 2015 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Surely you meant "integrable" not "non-integrable" here?

     
  6. Nov 13, 2015 #5
    :-) Ok thanks, I'll try to follow that notation in the future

    No, why do you say that? ##u = 1## is non-integrable on ##[0,+\infty[## since ##\int_0^x u(t) \ dt ## does not have a finite limit as ##x## tends to infinity.
     
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