I have the following definition:

$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$

From this, how can I get the definition of

$$\lim_{x\to p^-}=-\infty? $$

Try to formulate your collection of mathematical symbols in English and then try to understand what it says. Then, adapt it according to the other situation. Translate back into mathematical symbols after that.

Change "[tex]f(x)> \epsilon[/tex]" to "[tex]f(x)< -\epsilon[/tex]".  (With still [tex]\epsilon> 0[/tex].)