Asymptotes

Recall that if or or or , then the line is a vertical asymptote of the graph of a function .

Geometrically, the graph of gets as close as we like to the line if is sufficiently close to from the right or from the left, as shown in the animation below.

We say the line is an asymptote of the graph of a function if

or

Horizontal asymptotes

If or , then we say that is a horizontal asymptote the graph of .

Indeed, says that the value of gets as close as we like to if is sufficiently large. The animation below shows that the graph of gets as close as we like to the line if is sufficiently large.

The graph of has both coordinate axes as asymptotes.

What are the asymptotes of the graph of ?

It seems that . To verify this, let as approaches to , approaches . Thus

Since (why ?), we have .

Therefore, the graph of has an horizontal asymptote . Does the graph of have vertical asymptotes ?

Since for all nonzero , by the Squeeze Theorem, we see that . So the graph of have no vertical asymptote.

Here is the whole picture of the graph of .

What about the function ?

Since , which we have shown above, equals to . So the graph of has an horizontal asymptote .

And , so the graph of has no vertical asymptote either.

Here is how the graph of looks like, notice how the graph oscillates around the horizontal asymptote .

What are the asymptotes of the graph of ?

Due to round-up error, the graph might look funny if you keep zooming in.

In fact, = . This says that the graph of has no vertical asymptote. However,

=

Hence is a horizontal asymptote of the graph of .

Now let's take a look at the function . It is not hard to see that the graph of has a vertical asymptote , are there any other asymptotes ?

From the graph above, we see that the graph of looks very much like as the absolute value of gets large enough. Indeed, from the fact that

and ,

is also an asymptote of the graph of .