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Math Help - When does a limit not exist?

  1. #1
    Senior Member Twig's Avatar
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    When does a limit not exist?

    Hi

    Kinda of a fundamental question here.

    When does one say that a limit exists or not exist?

    For example)

     \displaystyle\lim_{x\to 0^{+}} \frac{1}{x} = \infty

    This limit exists and is equal to infinity?

    Is a typical case of a limit NOT existing  \displaystyle\lim_{x\to 0} \frac{|x|}{x} ? Because the limit is either negative one or plus one depending on if you approach zero from left or right.

    How can we really say that a limit 'equals' infinity? Infinity is not a number.

    Thx
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  2. #2
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    Quote Originally Posted by Twig View Post
    When does one say that a limit exists or not exist?
    For example)  \displaystyle\lim_{x\to 0^{+}} \frac{1}{x} = \infty
    This limit exists and is equal to infinity?
    Is a typical case of a limit NOT existing  \displaystyle\lim_{x\to 0} \frac{|x|}{x} ? Because the limit is either negative one or plus one depending on if you approach zero from left or right.
    How can we really say that a limit 'equals' infinity? Infinity is not a number.
    We can start a real argument right here and now with this question.
    The are many authors who write calculus textbooks that say
     \displaystyle\lim_{x\to 0^{+}} \frac{1}{x} does not exist.
    Some say the expression increases without bound.
    Still others have a vague way of extending the real numbers to include unbounded entities for which the symbol \pm\infty is used.
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  3. #3
    Senior Member Twig's Avatar
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    I see =)

    Yes I agree. I see many people saying that the limit equals infinity.
    My calculus teacher hated that.

    Thx
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  4. #4
    MHF Contributor

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    Quote Originally Posted by Twig View Post
    I see =)

    Yes I agree. I see many people saying that the limit equals infinity.
    My calculus teacher hated that.

    Thx
    Good for your teacher. I would say "the limit does not exist and is infinity!" Because saying a limit is "infinity" is just shorthand for saying the limit does not exist in particular way. Many Calculus texts will say that "1/x, as x goes to 2, converges to 1/2" but that "1/x, as x goes to 0, from above, diverges to infinity."
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  5. #5
    Senior Member pankaj's Avatar
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    Well, this is what I have read and believe it to be true.
    \lim_{x\to 0}\frac{1}{x^2}=\infty.
    The limit is said to exixt infinitely
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  6. #6
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by pankaj View Post
    Well, this is what I have read and believe it to be true.
    \lim_{x\to 0}\frac{1}{x^2}=\infty.
    The limit is said to exixt infinitely
    This limit does not exist because there is no value of epsilon that can be chosen such that

    |f(x)-L|<\epsilon whenever |x-a|<\delta

    I mean, how would you even approach the proof?

    |f(x)-\infty?|<? whenever |x-0|<\delta

    When we write \lim_{x\to{a}}f(x)=\infty, we are saying that the limit does not exist, but we are also saying why it doesn't exist.
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  7. #7
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Twig View Post

    Is a typical case of a limit NOT existing  \displaystyle\lim_{x\to 0} \frac{|x|}{x} ? Because the limit is either negative one or plus one depending on if you approach zero from left or right.
    Thx
    Yes. This limit DNE because \lim_{x\to{0^-}}\frac{|x|}{x}\neq\lim_{x\to{0^+}}\frac{|x|}{x}.
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  8. #8
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    Argueable

    I've always been taught that infinity is a concept, not a number. It just allows you to understand the long term behavior of a function. up to you
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