Infinite Limit Theorem

Theorem
Let $f$ be a real function of $x$ of the form


 * ${f\left({x}\right)} = \dfrac {g\left({x}\right)}{h\left({x}\right)}$

Further, let $g$ and $h$ be continuous on some open interval $\mathbb I$, where $c$ is a constant in $\mathbb I$.

If:
 * $\forall x \in \mathbb I, \ g(x) \ne 0$

And
 * $h(c) = 0$

And
 * $\forall x \in \mathbb I : x \ne c$, $h(x) \ne 0$

Then the limit will not exist and


 * $\displaystyle \lim_{x \to c ^+} f \left({x}\right) = +\infty$ or $-\infty$


 * $\displaystyle \lim_{x \to c ^-} f \left({x}\right) = +\infty$ or $-\infty$

Note
This theorem is absolutely not saying that $\dfrac c 0 = \infty$ when dealing with real numbers. Division by zero is undefined.