水平渐近线的极限

Just as we can visualize sequence convergence as points on the number line getting closer and closer to some number, we can use a similar visual tool to find what a function tends to at infinity.

To start, come up with some simple function of $x$ that converges at infinity.

My function was $f(x)=\frac{1}{x}$, so that's what we'll be working at.

OK, so $f(x)$ converges to $0$ as $x \to \infty$.

What does this look like on the graph? Press continue when you've thought about what the graph looks like.

When a function $f(x)$ approaches a horizontal line as $x \to \pm \infty$, then that line $y=a$ is said to be a horizontal asymptote of $f(x)$. This value of $a$ is the same as the limit of the function!

Note that the horizontal asymptotes needn't be the same in both the positive and negative direction. For example, consider $f(x)=\arctan(x)$.

Sketch its graph for yourself, and think about where the asymptotes are.

Here, $\lim_{x \to \infty} f(x) = \frac{\pi}{2}$ while $\lim_{x \to -\infty} f(x) = -\frac{\pi}{2}$. As you can see, the graph has two different horizontal asymptotes.

To conclude, try thinking about other convergent functions (like the one you came up with earlier!). What do their graphs look like? Where are their horizontal asymptotes?

(Video) Horizontal Asymptotes of Limits at Infinity

by PatrickJMT

Here's a video by patrickJMT showing you how to find the horizontal asymptote of a function whose limit approaches infinity.