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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Lab Day 2 -- Understanding $\lim{(x,y) \to (a,b)} f(x,y)$ geometrically}
\centerline{September 18, 2007}
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{\it Background and Goals}
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Last time in class we talked about what it should mean to say 
$$\lim_{{\bf x} \to {\bf a}} f({\bf x}) = L$$
for a function $f : {\bf R}^n \to {\bf R}$.  The limit
being $L$ means that
no matter how small we take $\varepsilon > 0$, it is
possible to find some small open ball 
$$B_\delta = \{{\bf x} \in {\bf R}^n\ |\ \|{\bf x} - {\bf a}\| < \delta\}$$
such that $f({\bf x}) \in (L - \varepsilon, L + \varepsilon)$ (or equivalently,
$|f({\bf x}) - L| < \varepsilon$)
for all ${\bf x} \in B_\delta$, except possibly for ${\bf x} = {\bf a}$. 

{\it Intuitively, when $n \ge 2$, we have lots of ways to let ${\bf x}$
approach ${\bf a}$ within any such open ball, and no matter how we do it, 
the values $f({\bf x})$ should be approaching $L$ as we get closer and
closer to ${\bf a}$.}

The goal of today's lab is to really ``dig in'' and understand
what this means in geometric terms by analyzing a fairly complicated
example where it is probably not immediately clear whether
the limit exists or not.  We will use Maple to draw several different
kind of plots and gather evidence.
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{\it The functions in question}
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The functions we will study are
$$f(x,y) = \cases{ x \sin(1/y) & if $y \ne 0$\cr
                      0 & if $y = 0$\cr}$$
and
$$g(x,y) = {x^4 y^4\over (x^2 + y^4)^3}.$$
The questions are:
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\centerline{\it Does $\lim_{(x,y) \to (0,0)}$ of each function
exist, and if so, what is it?}
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{\it Lab Questions}
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Before starting in enter the command
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\centerline{\tt with(plots);}
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to load the plots package.  
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I.  In this question, you will study the graph $y = f(x,y)$ as above.
Enter the following Maple command to 
define the function $f$ by name:
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\centerline{\tt f := (x,y) -> x*sin(1/y) + 1;}
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{\it Note:} We will ignore the second part of the definition in entering
the function into Maple.  This is OK as long as we don't end up 
asking Maple to evaluate $f$ at a point with $y = 0$ to draw
graphs.  This is avoided with the choices of ranges of $x$- and $y$-
values, and the grid options indicated.
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A)  (The first piece of evidence.)  Use Maple to generate the graph
$z = f(x,y)$, using the command
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\centerline{\tt plot3d(f(x,y), x=-5..5,y=-5..5,grid=[100,100]);}
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Does it look like there is some $z$-value $z = L$ such that {\it all the points} on 
the graph for $(x,y)$ near $(0,0)$ are close to level $L$?  Explain.
(Suggestion:  Use ``Boxed'' axes option and look at the scale on the $z$-axis.)
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B) (More evidence.) To confirm or contradict your ``hunch'' from part A, generate new graphs
by shrinking the ranges of $x$-and $y$-values as follows
\itemitem{1)} {\tt x = -1..1, y = -1..1}
\itemitem{2)} {\tt x = -0.1..0.1, y = -0.1..0.1}
\itemitem{3)} {\tt x = -0.01..0.01, y = -0.01..0.01}

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Use ``Boxed'' axes option and look at the scale on the $z$-axis.
For the smaller ranges here, you may see a $\times 10^{-3}$
printed out next to each coordinate axes.  This means that 
the tickmarks should all be multiplied by that factor.
Does this confirm or contradict what you said in part A?

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II.  Enter the following Maple command to 
define the function $g$ by name:
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\centerline{\tt g := (x,y) -> x\^{}4*y\^{}4/(x\^{}2 + y\^{}4)\^{}3;}
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A)  (The first piece of evidence.)  Use Maple to generate the graph
$z = g(x,y) = {x^4 y^4\over (x^2 + y^4)^3}$, using the command
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\centerline{\tt plot3d(g(x,y), x=-5..5,y=-5..5,grid=[60,60]);}
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Does it look like there is some $z$-value $z = L$ such that all the points on 
the graph for $(x,y)$ near $(0,0)$ are close to level $L$?  Explain.
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B) (The second piece of evidence.)  The previous question shows that 
the existence of a limit should be related in some way to the level
curves or contours of the function $g$.  In this part, you will draw
some of them and try to analyze the $g(x,y)$ using that information.
The level curve for the value $c$ is $g(x,y) = c$, or clearing denominators
$$x^4 y^4 = c(x^2 + y^4)^3.$$
(this form is slightly more reasonable for graphing than the 
direct equation $g(x,y) = c$.  
For example, the level curve for $c = 1/4$ can be plotted in Maple using
a new command called {\tt implicitplot} like this:
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\centerline{\tt implicitplot(x\^{}4y\^{}4 = (1/4)*(x\^{}2 + y\^{}4)\^{}3, 
x=-5..5,y=-5..5,grid=[60,60]);}
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Plot several of these level curves and try to understand how 
they relate to the 3d plot from part A.  Give the range of $c$ values 
that produces nonempty level curves and describe their shapes
of the level curves in words.  {\it Note:}  You can plot several 
of the level curves together on the same axes by putting a list 
of equations in [ ], separated by commas, in the {\tt implicitplot}
command.  See the Maple online help feature for more details
about this (and all Maple commands).
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C)  (``Not so fast!'')  In an example we looked at yesterday, we were
able to tell that (for a different function), $\lim_{(x,y) \to (0,0)} g(x,y)$
{\it did not} exist by {\it letting $(x,y) \to (0,0)$ along different lines
through $(0,0)$}.  What happens if you try that here?  (This computation can be done
by hand, or by asking Maple to evaluate
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\centerline{\tt g(t,m*t);}  
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for various values of $m$ you can supply, or even doing the 
same leaving $m$ as a symbolic variable.)  How does your answer here relate 
to the level curves you saw in part B?  Explain.   
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D)  (The final answer.)  Think in more detail about your pictures from 
part B.  What happens if you approach $(0,0)$ {\it along one of those curves}?
What does this say about $\lim_{(x,y) \to (0,0)} g(x,y)$.


\bye
