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:

$(1): \quad g \left({c}\right) \ne 0$

$(2): \quad h \left({c}\right) = 0$

$(3): \quad \forall x \in \mathbb I: x \ne c \implies h \left({x}\right) \ne 0$

then the limit as $x \to c$ 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 $-\infty$

Intuition

Consider the graph of the reciprocal function at the origin. Immediately to the right of the origin, for very small positive numbers, the function value gets very, very large. Similarly, immediately to the left of the origin, for very small (in magnitude) negative numbers, the function value gets very, very large (in magnitude), that is, very negative.

Note

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

Sources

• Roland E. Larson,  Robert P. Hostetler and Bruce H. Edwards: Calculus: 8th Edition (2005): $\S 1.5$