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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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    @Mercy This was before the edit was made to to $\epsilon=L/2$. I got it now.2012-09-27