formal definition of lim x->c+ f(x) = ∞

Question

how to write the formal definition of the following limits

$lim$ $x→c^+ $ $ f(x) = ∞$

and

$lim$ $x→c $ $ f(x) = ∞ $

Can I write it as $\forall\epsilon > 0, \exists\delta > 0 \thinspace s.t. \thinspace 0<|x-c|<\delta => |f(x) - \infty | < \epsilon$

Answer 1

No, because $|f(x)-\infty|$ does not mean anything.

Still, we want to capture $"f(x)$ getting closer and closer to $\infty"$ in some way. What this must mean is that $f(x)$ keeps getting larger, eventually surpassing any number. So we need it to be the case that for any number $K,$ we can make $f(x)$ larger than $K$ by making $x$ sufficiently close to $c.$ We can translate this to:

For any $K>0,$ there exists a $\delta >0$ such that $|x-c|< \delta \implies f(x)>K.$

This is a definition of $\lim_{x\to c} f(x) = \infty.$

Notice $K$ plays a role similar to $\epsilon$ here but is thought of as a large number rather than a small number in order to handle the fact that we're approaching $\infty$ rather than some real number.

I'll let you handle the one-sided case $\lim_{x\to c^+} f(x) = \infty.$