$\forall x \in R$ $g(x)\gt 0 $ and $\lim_{x \to \infty} \frac{f(x)}{g(x)}=L \gt0$, prove exists $N \in R$ such that $\forall x\gt N$ $f(x) \gt0$

Question

Given: $\forall x \in \mathbb R$ $g(x)\gt 0 $ and $\displaystyle \lim_{x \to + \infty} \frac{f(x)}{g(x)}=L \gt0$, prove that:

$\exists N \in \mathbb R$ such that $\forall x\gt N$ $f(x) \gt0$

So starting with the definition of the limit when $x$ approaches $+\infty$ we get

$ \forall \epsilon \gt 0, N \in \mathbb R$

$$x\gt N \Rightarrow \left| \frac{f(x)}{g(x)}-L \right| \lt \epsilon $$

if I pick $\epsilon=1$ I'm left with an inequality for $f(x)$ that leaves me worse off, forcing me to pick $L=1$ to make the inequality true, whereas I'm sure I want to prove this for every $L.$

Answer 1

If you take $\varepsilon = \dfrac{L}{2}$, then you have the existence of $N\in\mathbb R$ such that $$x> N \Rightarrow \frac{f(x)}{g(x)}>L-\frac{L}{2} = \frac{L}{2}>0$$ thus $$x> N \Rightarrow f(x) > \frac{L}{2}g(x)>0.$$

Answer 2

A formal proof requires $\epsilon, \delta$ type arguments, but a totally rigorous informal answer is as follows: if $f(x)/g(x) \to L$ and $L > 0$ then it is possible to ensure that all values of $f(x)/g(x)$ are as close to $L$ as we want by taking all suitable large values of $x$. But since $L > 0$ and if we get too close to $L$ then we must reach positive values. Thus $f(x)/g(x) > 0$ for all large values of $x$ and since $g(x) > 0$ we must have $f(x) > 0$ for all large values of $x$.