definition $\epsilon$ de $\limsup_{t\to+\infty}$

Question

Please what is the $\epsilon$ definition of this limit $$\limsup_{t\to+\infty}\frac{|f(t)|}{|t| |g(t)|}<+\infty$$

is it $$\forall \varepsilon>0, \exists A>0,\forall t\in\mathbb{R}, |t|>A\Rightarrow |f(t)|<\varepsilon |t| |g(t)| +\ell$$

thank you

Answer 1

As pointed out in the comments, not quite. Here's a definition. There exists an $l$ such that for every $\varepsilon > 0$ there exists $M$ such that $t > M$ implies $$ \frac{|f(t)|}{|t||g(t)|} < l + \varepsilon $$ and for infinitely many $t$, $$ \frac{|f(t)|}{|t||g(t)|} > l - \varepsilon $$

Answer 2

$$\forall \varepsilon>0, \exists B > 0, \forall A > B, \exists t>A,|t||g(t)|(\ell-\varepsilon) < |f(t)|< |t| |g(t)| (\ell + \epsilon)$$