In this section, I'll discuss the limit of a function as x goes to and . We'll see that this is related to *
horizontal asyptotes* of a graph.

It's natural to discuss * vertical asymptotes* as
well, and I'll explain how these are connected to values of x where
the limit of becomes infinite.

Let's start with an example. Here is the graph of :

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

Here's a rough definition. If the graph of approaches as you plug in
larger and larger *positive* values for x, then

Likewise, if the graph of approaches as you plug in larger and larger *negative*
values for x, then

As a numerical example, consider . If you set , you get

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

Here are the precise definitions. They're analogous to the definitions of ordinary limits.

* Definition.* means: For
every , there is a number M, such that:

(You can give a similar definition for .)

The definition says that I can make as close to L as I want, by making x sufficiently large.

As the picture shows, values of x greater than M produce values of that lie within of L.

* Example.* Prove that .

* Scratch work.* I'll start by working backwards
from to M.

(I can remove the absolute value bars, since means x will be large and positive.)

So

This suggests that I should take .

The reason for doing things this way is that you may not prove something by assuming what you want to prove. So the "working backward" part isn't by itself a valid proof: It is possible that some of the steps aren't reversible. You can ensure that everything works properly by writing the proof in the correct order, from assumptions to conclusion.

* The real proof.* Let . Take .

Then if , I have

Note that since , the last inequality implies . So

Dividing by in the first step is okay, because (so the inequality doesn't "flip"). Likewise, the second step is okay, because , so is positive, so I can add the absolute values.

Continuing, I have

This shows that .

Most of the properties of ordinary limits hold for limits as .

* Theorem.* (a)

(b) If k is a number,

(c)

(d) If , then

The statements mean that if the limits on the right side of the equation are defined, then the limits on the left sides are defined, and the two sides are equal.

* Proof.* I'll prove (a) by way of example. As in
most limit proofs, you discover what to do by working backward
("on scratch paper"). Then you write the "real
proof" forward. I'll omit the scratch work in this case.

A reminder about something before I start: I'll use the * Triangle Inequality*, which says that if p and q
are real numbers, then

Suppose that

I want to show that

Let .

Since , I can find a number M such that if , then

Since , I can find a number N such that if , then

Suppose that . This means that and , so both of the inequalities hold.

Hence, adding the inequalities, I get

(I used the Triangle Inequality in the " " step.) This proves that

Similar ideas are used in the proofs of (b), (c), and (d), though in some cases the algebra involved is a little trickier.

Here is a property that I'll use frequently.

* Proposition.* Let . Then

* Proof.* Let . I must find
a number M such that if and is defined, then

Set . Note that is defined and positive, since and . Suppose . Since M is positive, so is x, so is defined and positive.

I have

Hence, .

Is it true that

It is --- provided that is defined. What could go wrong? Suppose . Then is undefined, since is not defined if x is negative and means that x is taking on negative values. On the other hand,

Here are some examples of limits at and .

* Example.* (a) Compute .

(b) Compute .

(c) Compute .

(a) In limits at infinity involving powers of x, the rule of thumb is that the biggest powers dominate. In this case, the biggest powers on the top and bottom are the 's. Therefore, the limit in (a) behaves almost like

So 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 :

(b)

In this case, 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.

(c)

Here the on the bottom beats out the on the top.

Suppose that

I noted above that this means that the graph of approaches the line as you move to the right.

Likewise, suppose

This 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 .

Therefore, 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.

Alternatively, to think of it the other way,

So if x is negative (because ), I have .

Thus, 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.* (a) Compute

(b) Compute

(c) Compute

(a)

(b)

(c)

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

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

What are and ?

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 .

This is a large negative number, which suggests that the limit is .

In similar fashion,

Here's the graph:

* Example.* is undefined at . Does it have a
vertical asymptote at ?

The fact that a function is undefined at an isolated value does not imply that it has a vertical asymptote there. The graph of 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 other words,

In particular, the graph does not have a vertical asymptote at .

Copyright 2018 by Bruce Ikenaga