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: Prove or disprove: Justify your answer using the def or limit. Real analysis

  1. Apr 6, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose f: ℝ-{0} → ℝ has a positive limit L at zero. Then there exists m>0 such that if 0<|x|<m, then f(x)>0.

    2. Relevant equations
    The definition of the limit of a function at a point is: (already assuming f to be a function and c being a cluster point)
    A real number L is said to be a limit of f at c if, given any ε>0, there exists a δ>0 such that if x is an element of the domain and 0<|x-c|< δ, then |f(x)-L|<ε.

    3. The attempt at a solution
    I'm proving this true as:
    Spse the limit as x→0 of f = L (being a positive real number.
    This implies that given ε>0, there exists δ>0 such that 0<|x-0|< δ, then |f(x) - L|<ε.
    Let ε=L,
    By definition of the limit of a function at a point there exiss δ>0 such that for x in ℝ - {0}, 0<|x|<δ and |f(x) - L|<L if and only if 0<f(x)<2L.
    Thus taking m = δ, then for 0<|x|<m and f(x)>0.

    I'm sure it is sloppy, but it does make sense to me here. Please, constructive criticism is welcome! :)
  2. jcsd
  3. Apr 6, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    Makes sense to me.
  4. Apr 6, 2012 #3
    I agree this is actually quite well done.
  5. Apr 6, 2012 #4
    Thank you very much! Finally analysis is coming back around to itself so I can sort of see where things are going easier :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook