Here is the graph of :

The graph approaches the horizontal line as it goes out to the left and right. You write:

In general, to say that

means that the graph of approaches as you plug in larger and
larger *positive* values for x.

means that the graph of approaches as you plug in larger and
larger *negative* values for x.

For example, consider . If you set , you get

That's pretty close to 1, isn't it?

Let's look some examples of a limits at infinity.

* Example.*

In limits at infinity involving powers of x, the rule of thumb is that the biggest powers dominate. The limit above behaves almost like

because the 's on the top and bottom dominate. you expect the answer to be .

On way to see this formally is to divide the top and bottom by :

Now as ,

Hence,

Here's a picture of :

What else can happen?

because the on top beats out the puny on the bottom.

By the way, it would be correct to say this limit *diverges*.
However, it's more informative to say *how* it diverges. In
this case, the function becomes
large and negative, so you write for the limit.

On the other hand,

because the on the bottom beats out the on the top.

I noted above that

means that the graph of approaches the line as you move to the right, and

means that the graph of approaches the line as you move to the
left. In these situations, is a * horizontal
asymptote* for the graph of .

Not all graphs have horizontal asymptotes --- for example, goes to as and as . You can check for the presence of horizontal asymptotes by computing and and seeing if either is a number.

* Example.* Find the horizontal asymptotes (if
any) of .

Since

is a horizontal asymptote for the graph at and at . The graph is shown below:

* Example.* Find the horizontal asymptotes of
.

The limit at works without any surprises. The highest power on the top and the bottom is x (since looks like x), so divide the top and bottom by x:

However, the limit at is a little tricky! Here's the computation:

Where did that negative sign come from? Look at the bottom, which was
. x is going to , so x is taking
on *negative* values. Now is positive,
so is *negative*.

When you push the into the square root, you must
leave a negative sign outside. Otherwise, you'd have , a *positive* thing.

This is a case where it matters that x is going to , as opposed to . Here's the graph:

How do logarithms and exponentials behave as or ? The relevant facts are summarized below.

I've graphed (on the left) and (on the right) below; you can see that the pictures are consistent with the formulas above.

For example, the graph of goes downward asymptotically along the y-axis from the right. This confirms that .

Likewise, the graph of rises sharply as you go to the right; this confirms that .

Note that if in , the limits are reversed. Specifically,

* Example.*

Infinity can also appear in limits in connection with * vertical asymptotes*. I'll say that the graph of a
function has a * vertical asymptote* at
if at least one of the limits

is either or .

* Example.* The graph below has a vertical
asymptote at :

In this case,

In general, you might *suspect* the presence of a vertical
asymptote at an *isolated* value of x for which is
undefined. To * confirm* your suspicion, you need
to compute the left- and right-hand limits at the point.

* Example.* Locate the vertical asymptotes of
and sketch the graph near the
asymptotes.

is undefined at and at . I'll check for vertical asymptotes by computing the left- and right-hand limits at and at . I'll work through the first one carefully.

To see this, consider numbers close to 1 but to the right of 1. Then will be positive, while will be negative. For example, if , then while . All together, the fraction will be negative. But plugging into the fraction gives . Since the result is negative and infinite, it must be .

You can see numerical evidence for this by plugging (e.g.) into .

a large negative number.

In similar fashion,

Here's the graph:

* Example.* The fact that a function is
undefined at an isolated value does not imply that it has a vertical
asymptote there. For example, is
undefined at . The graph looks like this:

You can see this by noting that, for ,

Thus, the graph is the same as the graph of the line except at , where there's a hole. In particular, the graph does not have a vertical asymptote at .

Copyright 2005 by Bruce Ikenaga