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Math Help - Is this an infinite limit at infinity

  1. #1
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    Is this an infinite limit at infinity

    Ok so here is the limit:

    lim      (x^3-4x)/(7x)
    (x->∞)

    Now I know that the limit does not really exist, because the degree of the numerator is greater than the degree of the denominator. And this is the answer that they have on my handout.

    But I was wondering, would this be considered an infinite limit at infinity as in

    lim      (x^3-4x)/(7x)  =
    (x->∞)

    Is the above not true?

    And if not, how can I tell when the limit does not exist as above and when it should be infinity. I know that all polynomial functions have infinite limits at infinity. But there are rational limits that I know have infinite limits at infinity as well, such as:

    lim  (2x^2-4x)/(x+1)  =
    (x->∞)

    This one I know is true.

    I have a test in the morning on this, so anyone that could add some input that will help clarify this for me in the next 6 hours would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by CoryG89 View Post
    Ok so here is the limit:

    lim  (x^3-4x)/(7x)
    (x->∞)

    Now I know that the limit does not really exist, because the degree of the numerator is greater than the degree of the denominator. And this is the answer that they have on my handout.

    But I was wondering, would this be considered an infinite limit at infinity as in

    lim  (x^3-4x)/(7x) =
    (x->∞)

    Is the above not true?

    And if not, how can I tell when the limit does not exist as above and when it should be infinity. I know that all polynomial functions have infinite limits at infinity. But there are rational limits that I know have infinite limits at infinity as well, such as:

    lim  (2x^2-4x)/(x+1) =
    (x->∞)

    This one I know is true.

    I have a test in the morning on this, so anyone that could add some input that will help clarify this for me in the next 6 hours would be greatly appreciated.
    Dear CoryG89,

    Your first limit could be simpified like this,

    \lim_{x\rightarrow{\infty}}\frac{x^3-4x}{7x}=\frac{1}{7}\lim_{x\rightarrow{\infty}}x^2-\frac{4}{7}\lim_{x\rightarrow{\infty}}1=\frac{1}{7  }(\infty-4)=\infty
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  3. #3
    MHF Contributor

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    However, you should understand that " \infty" is not a real number- saying that a limit is " \infty" is just saying that the limit does not exist for a particular reason. Either "the limit does not exist" or "the limit is \infty" is correct, the latter being perhaps better as it says more.
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