Prove by the precise definition that if $\lim_{x→a} f(x) = ∞$, then $\lim_{x→a} (−f(x)) = −∞$.

Question

Prove by the precise definition that if $\lim_{x→a} f(x) = ∞$, then $\lim_{x→a} (−f(x)) = −∞$.

Let $M>0$, since $\lim_{x→a} f(x) = ∞$, there exists $δ>0$ such that $$ 0<|x-a|<δ\implies f(x)>M $$

This is what i know, but how do i then prove that the change to $-f(x)$ results in a $−∞$ limit.

Thanks.

Answer 1

You have to prove that for every $m<0$, there exists some $\delta$ such that if $$0<|x-a|<\delta\implies -f(x)<m$$

To do that, here's a simple hint:

What happens to the inequality $-f(x)<m$ if I multiply it by $-1$?