### Introducing Asymptotes With *Mathematica*

Tim Truitt & John Burnette

### •Intution versus Precision

#### •Intuitive Concepts

##### •(1) An asymptote is something you get really, really close to, but you never touch.

Not exactly, but it is a very common misconception. Consider the following function

It's very instructive to think about what the source of that misconception is.

##### •(2) If you "zoom out" far enough, this is what the function "acts like"

We happen to like this notion of "global behavior" a lot, but there are some drawbacks to this intuition. Consider what happens when you "zoom out" the graph of the following function

**Zoom-Movie**

It "looks like" this function becomes ever more closer to y=0. But if you "zoom out" you won't see the continuing variability.

**Another example**

**So we want to use the notion of "global behavior" but we also want to avoid the notion of "dominant term"**

##### •(3) You get vertical asymptotes whenever you divide by zero

Obviously we want our students to understand the limit process that creates a vertical asymptote.

<<Insert best "movie" to demonstrate the concept.>>

##### •(4) ... .unless the same factor is in the numerator, then you get a hole ...

In an ideal world, students would understand the difference between the graphs of

#### •Precise Definitions

##### •(1) Vertical asypmtotes

The function f(x) will have a vertical asymptote at x = a if and only if

##### •(2) Global behavior asymptotes (our nomination)

The polynomial P(x) is **A** (there could be two..) global behavior asymptote of f(x) if and only if

##### •(3) How technology (like *Mathematica*) changes what you can discuss in class

Suppose

Obviously, the rational expression of f[x] represents a polynomial division. In *Mathematica* you can rewrite this as a proper fraction using the *Apart* command.

However, we've learned today not to completely rely on a graph to decide if something is the global behavior, so we will use the built in ability of *Mathematica* to check.

So by our definition, it is *the* global behavior.

### •Using Technology for Demonstration

#### •Creating a MOVIE

It is our goal here to walk through the process of creating an animation which walks the students through the graph of a rational function. An animation is nothing more than a collection of Frames or "plots" which are shown one after the other. We need to start with an empty list and build it up.

What happens next is that we have a root, we'll have to make note of that by connecting it visually to the factor which creates it.

Next, we want to connect our function "turning negative" to the change of the sign of the factor of (x+3). We'll do this with color and hope that not too many of our students are colorblind.

Next, we want to connect our function's denominator "becoming zero" both to the existance of a vertical asymptote AND a change of sign.

And finally, we combine to make one large movie.

Converted by *Mathematica*
(March 15, 2005)