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Suppose f : D → R. Using only the defnition of limit of a function, show that if $\lim_{x\rightarrow x_0} f(x)=L>0$ then there is a number $d>0$ such that $f(x)≥L/2$ for all $x \in (x_0 −d,x_0 +d) \cap D$.

I' m not really sure where to go after writing down the definition of the limit for the assumption. Thanks in advance.

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Take $\varepsilon=\frac{L}{2}$ in the definition of limit and you are done. Of course I assume you learnt the $\varepsilon$--$\delta$ definition.

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I did learn the definition. Although even with taking $\epsilon=D/2$ I'm not seeing how this shows $f(x)\geq L/2$ –  tk2 Sep 27 '12 at 17:39
    
You can prove the existence of a $d$, satisfying your conditions. If you want to find the value, then you need other data regarding the function. –  Phani Raj Sep 27 '12 at 18:20
    
@tkrm You don't see it because you haven't tried it! Btw it should be $\varepsilon=L/2$, not $\varepsilon=D/2$. –  Mercy Sep 27 '12 at 19:09
    
@Mercy This was before the edit was made to to $\epsilon=L/2$. I got it now. –  tk2 Sep 27 '12 at 19:19

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