When do you know you have an infinite limit, indicating a
vertical asymptote?
Consider a limit $\displaystyle\lim_{x\to a}f(x)$.
Plug in $a$ for $x$ in $f$. If you get an answer with a
non-zero numerator and a zero in the denominator, there is a vertical asymptote at $x=a$, and thus the value of the limit will be $\infty$
or $-\infty$, or the limit will not exist.
To see which of those three possibilities occurs, evaluate the
limit from the left and from the right. I.e. evaluate $\displaystyle\lim_{x\to a^-}f(x)$ and
this will be either $\infty$ or
$-\infty$. Now evaluate $\displaystyle\lim_{x\to
a^+}f(x)$ and this will be either
$\infty$ or $-\infty$.
These limits indicate the behavior of the graph of $f$ as
$x\to a$ from the left, and as $x\to a$ from the right.
If, from both directions, the limit is $\infty$, then
the limit is $\infty$.
Similarly, if, from both directions, the limit is
$-\infty$, then the limit is $-\infty$.
If one of the limits is $\infty$ and the other is
$-\infty$, the limit does not exist.
You can use this technique on the questions that follow the
videos.
A special case: vertical asymptotes for rational functions
Rational functions often have vertical asymptotes when the
denominator goes to zero (and the numerator doesn't), such as
$f(x) =\dfrac{3}{x-4}$ or $f(x)=\dfrac{x^2-1}{x^2-4}$. We
explore rational function vertical asymptotes here.