So far we have studied limits as $x \to a^+$, $x \to a^-$ and $x
\to a$. Now we will consider what happens as ''$x \to \infty$'' or
''$x \to -\infty$". What does that mean?

$\displaystyle{\lim_{x \to \infty}}$ describes what happens when $x$
grows without bound in the positive direction. The word ''infinity''
comes from the Latin "infinitas", which stands for "without end" (in=not, finis=end). Imagine taking
bigger and bigger values of $x$, like a hundred, a thousand, a
million, a billion, and so on, and seeing what $f(x)$ does. For
instance, the statement $\displaystyle{\lim_{x \to \infty}} f(x)=7$ means
that, as $x$ grows larger and larger (and positive), $f(x)$ is closer and closer to 7.

$\displaystyle{\lim_{x \to -\infty}}$ is the same thing, only in the
negative direction. Look at $x$ being minus a million, minus a billion,
minus a trillion, etc.

If $\displaystyle{\lim_{x \to \infty} f(x) = 3}$, then the graph of
$y=f(x)$ will be very close to the horizontal line $y=3$ when $x$ is
large and positive. We call the line $y=3$ a horizontal asymptote.

Likewise, if $\displaystyle{\lim_{x \to -\infty} f(x) = 7}$,
then $f(x)$ is close to 7 whenever $x$ is large and negative,
so the graph of $y=f(x)$ is very close to the horizontal asymptote $y=7$.

Horizontal Asymptotes

Definition:
The line \(y=L\) is called a horizontal asymptote for \(y=f(x)\) if and only if
\[
\lim_{x\to\infty}f(x)=L, \quad \text{ or }\quad \lim_{x\to-\infty}f(x)=L
\]

For instance, the graph on the left has both $y=\pi/2$ and $y=-\pi/2$ as horizontal asymptotes. The one on the right has horizontal asymptotes $y=\pm 4$. (Can a function have more than two horizontal asymptotes?)