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?
$\displaystyle{\lim_{x \to \infty}f(x)}$ describes what
happens to $f$ 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, $f(x)$ is closer and closer to 7.
We call the line $y=7$ a horizontal
asymptote of $f$, since as $x$ grows larger and
larger, $f(x)$ starts looking like the line $y=7$.
$\displaystyle{\lim_{x \to -\infty}f(x)}$ is similar, but 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 negative. Then the line $y=3$
is a horizontal asymptote of $f$.
Horizontal Asymptotes
Definition:
The line \(y=L\) is called a horizontal asymptote
for \(y=f(x)\) if and only if \[ \lim_{x\to\infty}f(x)=L,
\quad \text{ or }\quad \lim_{x\to-\infty}f(x)=L \]
Can a function have more than two horizontal asymptotes?
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 $y=\pm 4$.
