CCD.DLC.Example.3. A Vertical Asymptote Discontinuity ("Blowing Up").
Functions that have a definition that specify a value at one number when
an algebraic rule fails sometimes fail to be continuous at that number
only because there is no possible limit of the
function at that number because the values of $f(x)$ are larger (or
smaller) than any particular number $M$. Graphically this can occur at a
number $a$ where there is a vertical asymptote.
In other words the function fails to be
continuous because there is no number $L$ where
$\lim_{x \to a} f(x) = L $.
Such examples are commonly used to illustrate functions that are not
continuous
at a single point because
$\lim_{x \to a} f(x)$ does not exist.
Consider $f(x) = \frac 1{x-1}$ when $x \ne 1$ and $f(1)=2$.
Then $\lim_{x \to 1} f(x) = \lim_{x \to 1}\frac 1{x-1} \ne L$ for any value of $L$
so $f$ is not continuous at $x=1$.
In the GeoGebra sketch below with
both mapping diagram and graph to visualize the failure of the limit of $f$ at $x=1$.
Martin Flashman
Created with GeoGebra
Notice how the points on the graph are paired with the points
on the mapping diagram.