prove the positive limit of a function

Question

I am reading limit and continuity in the analysis. And stuck on a problem. Hope you guys will help me out. If $\lim_{x\to a} f(x)=L>0$, then there is a $\delta>0$ such that $f(x)>0$ when $0<|x-a|<\delta$. Thank you in advance.

Answer 1

Hint: What happens if you take $\varepsilon=L$?

Answer 2

$\lim\limits_{x\to a} f(x) = L \iff \forall \varepsilon > 0, \, \exists \delta>0$ such that $|f(x) - L|<\varepsilon$ when $0<|x-a|<\delta$

In particular, if $\varepsilon = \frac{L}{2}$ then $\exists \delta>0$ such that when $0<|x-a|<\delta$ :

$$|f(x) - L|<\varepsilon = \frac{L}{2} $$

So

$$-\frac{L}{2}<f(x) - L < \frac{L}{2}$$

which implies

$$0<\frac{L}{2}<f(x)$$

because $L>0$