0
$\begingroup$

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.

1 Answers 1

0

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

  • 0
    I did learn the definition. Although even with taking $\epsilon=D/2$ I'm not seeing how this shows $f(x)\geq L/2$2012-09-27
  • 0
    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.2012-09-27
  • 0
    @tkrm You don't see it because you haven't tried it! Btw it should be $\varepsilon=L/2$, not $\varepsilon=D/2$.2012-09-27
  • 0
    @Mercy This was before the edit was made to to $\epsilon=L/2$. I got it now.2012-09-27