Vertical Asymptotes (Redux)
What happens if a function $f(x)$ grows (or decreases) without bound
as $x$ approaches some value?
- If $\displaystyle{\lim_{x \to a^+} f(x) = \infty}$, then
$f(x)$ is large and positive
whenever $x$ is slightly greater than $a$.
- This means that the graph $y=f(x)$ is very close to the
vertical line $x=a$. We call this vertical line a vertical asymptote (you
might want to review vertical asymptotes).
- The same sort of thing happens if $\displaystyle{\lim_{x \to
a^+} f(x) = -\infty}$, or if $\displaystyle{\lim_{x \to a^-}
f(x) = \pm \infty}$. In all of these cases we have a vertical
asymptote at $x=a$.
- We have seen that $f(x)=\frac{1}{x}$ has a vertical
asymptote. Where is it?