What does a horizontal asymptote represent?

Sep 4, 2014

A horizontal asymptote is a fixed value that a function approaches as $x$ becomes very large in either the positive or negative direction. That is, for a function $f \left(x\right)$, the horizontal asymptote will be equal to ${\lim}_{x \to \pm \infty} f \left(x\right)$.

As the size of $x$ increases to very large values (i.e. approaches $\infty$), functions behave in different ways. Some simply get bigger and bigger forever (e.g. ${x}^{2}$). Others oscillate up and down (e.g. $\sin x$). But others get closer and closer to a particular constant value. For instance, consider the hyperbola $y = \frac{1}{x}$.

When we input very large values of $x$ (think of $x = 1000$, $x = 1000000$, $x = 1000000000$ etc.), the value of $\frac{1}{x}$ becomes very small. It gets very close to $0$. And if we plot points to see this visually, we find that the graph of $y = \frac{1}{x}$ approaches the line $y = 0$. We can observe the same effect when putting in very large negative values of $x$, too. So we say that $y = 0$ is the horizontal asymptote of $y = \frac{1}{x}$.

If we shifted the entire graph upward, say $y = \frac{1}{x} + 5$, then we would find that the graph approaches the line $y = 5$ instead. That's because ${\lim}_{x \to \pm \infty} \frac{1}{x} = 0$ and so all that's "left over" when considering the horizontal asymptote is the $5$.

A couple of important notes. Firstly, even though the graph approaches the asymptote, it will never get there. There is no value of $x$ we can input into $y = \frac{1}{x}$ that will make $y$ actually equal to $0$! So even though the graph approaches a certain value (like $x = 0$), it's critical to understand that "asymptotic behaviour" means you never actually get there.

Secondly, horizontal asymptotes tell you how a graph behaves at its extremities (i.e. as $x \to \pm \infty$). It doesn't tell you anything about how the graph behaves "in the middle", near the $y$-axis. For instance, going back to our example of $y = \frac{1}{x}$, the graph has nothing to do with $y = 0$ for small values of $x$. In some instances, such as $y = \frac{x}{{x}^{2} + 1}$, the graph can even cross the horizontal asymptote. This is a critical difference between horizontal and vertical asymptotes! (Vertical asymptotes are a different thing entirely, even though they look very similar.)