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

f[x_] := Sin[x]/x

Plot[

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

g[x_] := (3 x)/(x + 1) + Sin[x]

Zoom-Movie

For[a = 10, a < 200, a = a + 25, Plot[g[x], {x, -a, a}, PlotRange -> {-a, a}]]

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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

g[x_] := (x - 50)^2 + 250

Plot[g[x], {x, 0, 100}, PlotRange -> {-200, 2000}]

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-Graphics -

Plot[g[x], {x, -100, 200}, PlotRange -> {-500, 5000}]

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Plot[g[x], {x, -1000, 2000}, PlotRange -> {-1000, 1000000}]

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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

y = (f (x) (x - a))/(x - a)             ... = (f (x) (x - a))/(x - a)^2        (still has a vertical asymptote)

•Precise Definitions

•(1)  Vertical asypmtotes

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

Underscript[lim, x -> a^+] f (x) = ± ∞       &nb ... ;        Underscript[lim, x -> a^-] f (x) = ± ∞

•(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

Underscript[lim, x -> ∞^+] (f (x) - P (x)) = 0       &n ...         Underscript[lim, x -> ∞^-] (f (x) - P (x)) = 0

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

Suppose

f[x_] := (3 x^5 - 4 x^3 + 2 x + 12)/(x^3 - 2 x^2 + x - 4)

Plot[f[x], {x, -6, 7}, PlotRange -> {{-6, 7}, {-200, 300}}]

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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.  

Apart[f[x]]

5 + 6 x + 3 x^2 + (32 + 21 x + 16 x^2)/(-4 + x - 2 x^2 + x^3)

Plot[{f[x], 3 x^2 + 6 x + 5}, {x, -6, 7}, PlotRange -> {{-6, 7}, {-200, 300}}, PlotStyle -> {RGBColor[0, 0, 0], RGBColor[0, 1, 0]}]

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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.

Limit[f[x] - (3 x^2 + 6 x + 5), x -> ∞]

0

Limit[f[x] - (3 x^2 + 6 x + 5), x -> -∞]

0


So by our definition, it is the global behavior.

•Using Technology for Demonstration

•Creating a MOVIE

R[x_] := ((x - 2)^2 (x + 3))/(x + 1)

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.

For[a = -6, a < -3, a = a + .1, Show[{Plot[R[x], {x, -6, a}, PlotRange -> {{-6, 6}, {-6 ... GBColor[0, 0, 0], PointSize[0.03], Point[{a, R[a]}]}]}, DisplayFunction -> $DisplayFunction] ]

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.

Show[{Plot[R[x], {x, -6, -3}, PlotRange -> {{-6, 6}, {-60, 60}}, PlotLabel -> "\!χΙχΙ ... BColor[0, 1, 0], PointSize[0.03], Point[{-3, R[-3]}]}]}, DisplayFunction -> $DisplayFunction]

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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.

For[a = -3, a < -1, a = a + .1, Show[{Plot[R[x], {x, -6, -3}, PlotRange -> {{-6, 6}, {- ... ; RGBColor[1, 0, 0], DisplayFunction -> Identity] }, DisplayFunction -> $DisplayFunction] ]

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

Show[{Plot[R[x], {x, -6, -3}, PlotRange -> {{-6, 6}, {-60, 60}}, PlotLabel -> "\!χΙχΙ ... t; RGBColor[1, 0, 0], DisplayFunction -> Identity] }, DisplayFunction -> $DisplayFunction]

-Graphics -

For[a = -1, a < 2, a = a + .1, Show[{Plot[R[x], {x, -6, -3}, PlotRange -> {{-6, 6}, {-60 ... ; RGBColor[0, 0, 0], DisplayFunction -> Identity] }, DisplayFunction -> $DisplayFunction]]

Show[{Plot[R[x], {x, -6, -3}, PlotRange -> {{-6, 6}, {-60, 60}}, PlotLabel -> "\!χΙχΙ ... gt; RGBColor[0, 0, 0], DisplayFunction -> Identity] }, DisplayFunction -> $DisplayFunction]

For[a = 2, a < 6, a = a + .1, Show[{Plot[R[x], {x, -6, -3}, PlotRange -> {{-6, 6}, {-60, ... t; RGBColor[0, 0, 0], DisplayFunction -> Identity] }, DisplayFunction -> $DisplayFunction]]

And finally, we combine to make one large movie.

Converted by Mathematica  (March 15, 2005)