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Math Help - Limit Quotient Theorem Proof

  1. #1
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    Limit Quotient Theorem Proof

    Hello everyone. Can someone show me how to do this proof please for my Real Analysis class? Proofs have always been a problem for me. I cant seem to get them to flow so any tips would be greatly appreciated.

    Prove lim x->a f(x)/g(x) = L/M
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  2. #2
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    If I were you, I would go to a basic calculus textbook. In any such text you find a proof for a similar statement.
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  3. #3
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    Thanks Plato, that helped some. I forgot to mention it is supposed to be an epsilon delta proof.
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  4. #4
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    Here are the basics.
    \left| {\frac{{f(x)}}<br />
{{g(x)}} - \frac{M}<br />
{N}} \right| = \left| {\frac{{f(x)N - g(x)M}}<br />
{{g(x)N}}} \right| \leqslant \left| {\frac{{f(x)N - NM}}<br />
{{g(x)N}}} \right| + \left| {\frac{{NM - g(x)M}}<br />
{{g(x)N}}} \right|
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  5. #5
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    That helped more. Thanks again Plato. I'll get to work
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  6. #6
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    I've dragged this problem up again because I never really was able to finish it and I have a final next week and have a feeling something like this may be on it...

    I need to do it using episilon delta definitions so it starts something like this

    If 0  < \mid x - a \mid < \delta_1 \Rightarrow \mid f(x) - M \mid \leq \frac{\varepsilon}{2} = \varepsilon_1
    and 0 < \mid x - a \mid < \delta_2 \Rightarrow \mid f(x) - N \mid \leq \frac{\varepsilon}{2} = \varepsilon_2

    Then by algebra... how Plato started it...



    Now this is where I keep getting lost... I've spent many hours trying to get it where I can find \varepsilon_1 and \varepsilon_2. I always seem to end up with the g(x) somewhere and not able to get it so its all constants and substituted with  \varepsilon 's.
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