proofs with limits!!!

[rikaminami]

please help me! I've never really been good at limits and I'm totally lost with this one:

Given that lim g(x)=L, whereL >0, prove that there exists an open interval
x :arrow: c
(a,b) containing c such that g(x)>0 for all x cannot equal c in (a,b).

Please and thanks so much!

 

[pka]

Given that \(\displaystyle \lim _{x \to c} f(x) = L > 0\), use the \(\displaystyle \varepsilon \;\delta\) definition of limit.

\(\displaystyle \L \left( {\forall \varepsilon > 0} \right)\left( {\exists \delta > 0} \right)\left[ {0 < \left| {x - c} \right| < \delta\, \Rightarrow \left| {f(x) - L} \right| < \varepsilon } \right]\).

In that definition, take the particular case of:
\(\displaystyle \L \varepsilon = \frac{L}{2}\quad \Rightarrow \quad - \frac{L}{2} < f(x) - L < \frac{L}{2}\)
Now just add L to all the parts.