# Thread: Limit Quotient Theorem Proof

1. ## 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

2. If I were you, I would go to a basic calculus textbook. In any such text you find a proof for a similar statement.

3. Thanks Plato, that helped some. I forgot to mention it is supposed to be an epsilon delta proof.

4. Here are the basics.
$\left| {\frac{{f(x)}}
{{g(x)}} - \frac{M}
{N}} \right| = \left| {\frac{{f(x)N - g(x)M}}
{{g(x)N}}} \right| \leqslant \left| {\frac{{f(x)N - NM}}
{{g(x)N}}} \right| + \left| {\frac{{NM - g(x)M}}
{{g(x)N}}} \right|$

5. That helped more. Thanks again Plato. I'll get to work

6. 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.