Infinite Limits
The statement $$\lim_{x \to a} f(x) =
\infty$$ tells us that whenever $x$ is close to (but not
equal to) $a$, $f(x)$ is a large positive number. A limit with
a value of $\infty$ means that as $x$ gets closer and closer to $a$,
$f(x)$ gets bigger and bigger; it increases
without bound. Likewise, the statement $$\lim_{x \to a} f(x) = \infty$$ tells us
that whenever $x$ is close to $a$, $f(x)$ is a large negative
number, and as $x$ gets closer and closer to $a$, the value of
$f(x)$ decreases without bound.
Warning: when we say a
limit $=\infty$, technically the limit doesn't exist.
$\displaystyle\lim_{x\to a}f(x)=L$ makes sense (technically) only
if $L$ is a number. $\infty$ is not a number! (The
word "infinity" literally means without
end.) If the limit is $+ \infty$, then the
function increases without end. If the limit is $\infty$, it
decreases without end. We say a limit is equal to
$\pm\infty$ just to indicate this increase or decrease, which is
more information than we would get if we simply said the limit
doesn't exist.
Vertical Asymptotes
Definition: The line $x=a$ is a vertical
asymptote of a function $f$ if the limit of
$f$ as $x\to a$ from the left and/or right is
$\pm\infty$. This means at least one of the
following is true:
$$\lim_{x \rightarrow a^+}f(x) =
\infty,$$ 
$$\lim_{x \rightarrow a^}f(x) =
\infty,$$ 
$$\lim_{x \rightarrow a^+}f(x) =
\infty,$$ 
$$\lim_{x \rightarrow a^}f(x) =
\infty.$$ 

Here are some examples of graphs with one or more vertical
asymptotes.
DO: Find all vertical asymptotes in the
following graphs.
