Limits at Infinity and Horizontal Asymptotes

Limits at Infinity

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

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?)