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Math Help - Limit Law Questions

  1. #1
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    Limit Law Questions

    Hi everyone, I need some help with two limit law questions.

    1. Is it possible for the limit of f(x)*g(x) as x-->a to exist even if lim f(x) as x-->a and lim g(x) x-->a do not exist?

    2. Is it possible for the limit of f(x)+g(x) as x-->a to exist even if lim f(x) as x-->a and lim g(x) as x-->a do not exist?

    I know the first one is true, but I can't think of an example. Could someone please explain if either question is true or false and give an example that justifies their reason?

    Thanks.
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  2. #2
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    Quote Originally Posted by amma0913 View Post
    Hi everyone, I need some help with two limit law questions.

    1. Is it possible for the limit of f(x)*g(x) as x-->a to exist even if lim f(x) as x-->a and lim g(x) x-->a do not exist?
    I cannot think of an example where both limits do not exist by limit of f(x)g(x) does but if f(x)= x, g(x)= 1/x, the limit of g(x), as x goes to 0, does not exist while f(x)g(x)= x(1/x)= 1 so its limit is 1.

    2. Is it possible for the limit of f(x)+g(x) as x-->a to exist even if lim f(x) as x-->a and lim g(x) as x-->a do not exist?
    Look at f(x)= 1/x, g(x)= -1/x, as x goes to 0.

    I know the first one is true, but I can't think of an example. Could someone please explain if either question is true or false and give an example that justifies their reason?

    Thanks.
    Last edited by mr fantastic; September 17th 2009 at 05:54 AM. Reason: Added a close quote tag to the second quote.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by amma0913 View Post
    Hi everyone, I need some help with two limit law questions.

    1. Is it possible for the limit of f(x)*g(x) as x-->a to exist even if lim f(x) as x-->a and lim g(x) x-->a do not exist?

    2. Is it possible for the limit of f(x)+g(x) as x-->a to exist even if lim f(x) as x-->a and lim g(x) as x-->a do not exist?

    I know the first one is true, but I can't think of an example. Could someone please explain if either question is true or false and give an example that justifies their reason?

    Thanks.
    Let:

     <br />
f(x)= \begin{cases} 0 & x \text{ rational } \\ 1 & x \text{ irrational }<br />
\end{cases}<br />

    and:

     <br />
g(x)= \begin{cases} 1 & x \text{ rational } \\ 0 & x \text{ irrational }<br />
\end{cases}<br />

    CB
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