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: Improper integral convergence and implications of infinite limits

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Let f be a continuous function on [1,∞) such that [itex]\lim_{x\rightarrow ∞}[/itex]f(x)=α. Show that if the integral [itex]\int^{∞}_{1} f(x)dx[/itex] converges, then α must be 0.

    2. Relevant equations
    Definition of an Improper Integral
    Let f be a continuous function on an interval [a,∞). then we define [itex]\int^{∞}_{a} f(x)dx[/itex] to be [itex]\lim_{b \rightarrow ∞}\int^{b}_{a} f(x)dx[/itex] if this limit exists. In this case, the integral is said to be convergent.

    3. The attempt at a solution

    My attempt at this solution was first to start with the epsilon-delta version of a limit of a function. That is, since [itex]\lim_{x\rightarrow ∞}[/itex]f(x)=α, then [itex]\forall \epsilon[/itex]>0 with [itex]x_{0}\in [/itex] [1,∞) [itex]\exists~ \delta[/itex]>0 s.t. |f(x) - α | whenever |x-[itex]x_{0}[/itex]|<[itex]\delta[/itex]. Then, I assumed that α was not equal to zero. From there I split up my inequality so that

    -ε<f(x) - α< ε [itex]\Rightarrow[/itex] α - ε<f(x)< ε + α [itex]\Rightarrow[/itex] [itex]\int^{∞}_{1}[/itex](α - ε)dx < [itex]\int^{∞}_{1} f(x)dx[/itex] < [itex]\int^{∞}_{1} (α + ε)dx[/itex]

    From here, I am getting stuck in my logic and I think that perhaps this is not the route I want to go. I would appreciate it if I could be offered any hints on where to go from here, or on whether or not I should start over. Thanks!
    Last edited by a moderator: Apr 9, 2012
  2. jcsd
  3. Apr 9, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hi Szichedelic! :smile:
    No, for ∞, your δ is a number "near ∞" …

    we usually use N instead, and we look for an N such that x > N => |f(x) - α| < ε :wink:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook