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Homework Help: Limits with the precise definition of a limit

  1. Nov 4, 2014 #1
    • Member warned about not including a solution effort
    1. The problem statement, all variables and given/known data
    Suppose that limit x-> a f(x)= infinity and limit x-> a g(x) = c, where c is a real number. Prove each statement.
    (a) lim x-> a [f(x) + g(x)] = infinity
    (b) lim x-> a [f(x)g(x)] = infinity if c > 0
    (c) lim x-> a [f(x)g(x)] = negative infinity if c < 0
    2. Relevant equations
    The limit laws would seem relevant, but the f(x) limit goes to infinity.

    3. The attempt at a solution
    I'm completely lost on how to start this problem. How would I prove something like the limit laws using the precise definition of a limit when one of the limits don't exist?
  2. jcsd
  3. Nov 4, 2014 #2


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    Staff Emeritus
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    When you say that
    $$\lim_{x \to a} f(x) = \infty,$$ it's not exactly the same as saying the limit doesn't exist. For example, the limit
    $$\lim_{x \to 0} \sin \frac{1}{x}$$ doesn't exist, but you wouldn't say it's equal to infinity either. So what precisely does it mean when you write a limit equals infinity?
  4. Nov 4, 2014 #3


    Staff: Mentor

    Your textbook should have the precise definition of a limit when the function is unbounded. It doesn't use ##\delta## and ##\epsilon## as the normal limit does, but instead uses M and ##\delta##, where M is a large number.
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