A related question to this question, I am wondering $\lim_{x\to a}{f^\prime(x)}=+\infty,$ what can be concluded about $f(a)$? Does this invalidate that $f(x)$ is not continuous at $a$ because of the non-existence of $f^\prime(a)$? Does this condition also imply maybe $\lim_{x\to a}{f(x)}=+\infty?$
I would think $\lim_{x\to a}{f(x)}=+\infty,$ because here $f(a+h)-f(a)$ can be arbitrarily large no matter how small $h$ is.
EDIT
Okay, I see where I got it wrong. Even though $\lim_{x\to a}{f^\prime(x)}=+\infty,$ it does not mean $f(a+h)-f(a)$ is arbitrarily large, because an $\infty$ times an infinitesimal quantity may not be determinate.
I just wonder another related question: given $\lim_{x\to a}{f(x)}=+\infty,$ what can be concluded to $f^\prime(a)$? Can it be finite or non-existent? How about also when $a=\infty$ in this case.