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?

• $\displaystyle{\lim_{x \to \infty}f(x)}$ 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

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 $y=\pm 4$.