In calculus, when we encounter limits of the form “A/0” where A is non-zero, this is not an indeterminate form and by observing the functions, most of the time each side blows up to $+ \infty$ or $-\infty$
Is this always the case, ie. if $\lim_{x\to a} f(x) = k$, $k$ non-zero, and $\lim_{x\to a} g(x) = 0$ is $\lim_{x\to a} \lvert \frac{f(x)}{g(x)}\rvert = \infty$, where we take absolute value because we only care about the magnitude?
If so, how do we prove that limits in that form always eventually grow without bound and not just purely oscillating in some bounded amount? If not, what are counterexamples, and under what conditions where limits of that form have its magnitude blow up to infinity?