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Is a Real funcion with a Limit Bounded?

  1. Oct 2, 2011 #1
    Hi, just a quick question.

    Let f be real function s.t. the limit of f as x approaches a equals L.

    Is f bounded?
    i.e. is it sufficient to assume a function is bounded if it has a limit.

    Thanks to all who may reply.
  2. jcsd
  3. Oct 2, 2011 #2
    Yes it is. Use the definition of limit and choose your epsilon to be a certain constant (say 1). Then work your magic.
  4. Oct 2, 2011 #3
    Rather, it is bounded on some open set containing a.

    It might go off to infinity elsewhere; you just know that it doesn't go off to infinity right at that point.
  5. Oct 2, 2011 #4


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    There's not enough information to tell. The only x value that you can say f is bounded is x = a. For every other x value, we have no idea if there exists an x asymptote. Like the previous poster suggested, we only know that it's bounded on some open interval containing a.
  6. Oct 2, 2011 #5
    Just take f(x) = x. The limit of f as x -> a is defined and finite for every a, but f is not bounded.
  7. Oct 2, 2011 #6


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    I noticed a mistake in my previous post. f(x) isn't guaranteed to even exist for x = a. However, f(x) is still bounded in some neighborhood of x = a, since the limit exists there.
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