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What is limx-->0 f(x) when lim x-->0 g(x)=0?

  1. Oct 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Let |f(x)|≤g(x) for all x∈ Mf∩Mg.
    What is limx-->0 f(x) when lim x-->0 g(x)=0?
    What is limx-->0 f(x) when lim x-->0 g(x)=3?

    3. The attempt at a solution
    Well, given that|f(x)|≤g(x), lim x-->0 g(x)=0 intuitively implies to me that | limx-->0 f(x) |≤0
    therefore | limx-->0 f(x) |=0 --> limx-->0 f(x)=0

    AND when lim x-->0 g(x)=3

    |limx-->0 f(x)|≤3 ⇔ -3 ≤limx-->0 f(x)≤3

    Is my reasoning correct?
  2. jcsd
  3. Oct 4, 2015 #2


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    Homework Helper

    Can you turn your intuition into a rigorous proof that [itex]\lim_{x \to 0} f(x) = 0[/itex] in this case?

    Must [itex]\lim_{x \to 0} f(x)[/itex] necessarily exist when [itex]\lim_{x \to 0} g(x) > 0[/itex]? Consider the case [tex]
    f(x) = \begin{cases} -1 & x < 0, \\ 0 & x = 0, \\ 1 & x > 0, \end{cases}\qquad g(x) = 3.[/tex]
  4. Oct 4, 2015 #3

    No. So in that case we cannot say anythin about [itex]\lim_{x \to 0} f(x) [/itex] since it won't necessarily even exist?

    However if it exists then -3 ≤limx-->0 f(x)≤3
  5. Oct 5, 2015 #4
    ahh there is a small typo in the problem statement.. it should be x approaches a, not 0
  6. Oct 5, 2015 #5
    I used the squeeze theorem to prove that limx→0f(x)=0
  7. Oct 5, 2015 #6


    Staff: Mentor

    Isn't it the limit as x approaches a?

    So for the second part, if ##|f(x)| \le g(x)## and ##\lim_{x \to a} g(x) = 3##, is it necessarily true that ##\lim_{x \to a} f(x) = 0##?
  8. Oct 5, 2015 #7
    Yes it is, I'm sorry

    No, because limx→a f(x) won't necessarily exist when lim x-->a g(x)=3.
  9. Oct 5, 2015 #8


    Staff: Mentor

    OK, I wasn't sure whether you were using the squeeze theorem for the second part.
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