Calculus Animations,Graphics and Lecture Notes

Visualizing Limits of Functions of 2 Variables

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Visualizing Limits of Functions of 2 Variables
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The basic idea for limits of functions of one variable is that given any ε we can find a δ

such that if  we are within  δ of x = a in the domain then f(x) lies between the lines

y = L - ε and y = L + ε  . Formally  | f(x) - L| < ε  whenever | x-a | <  δ.

But what about a function of 2 variables?  I'll formalize the language later but for now we'll consider the following 3 animations

which illustrate qualitatively what limits of functions of 2 variables are all about.

The first difference when we talk about functions of 2 variables is that instead of considering an interval containing x = a in the domain

of the form a - δ < x < a + δ

We talk about a circle centered at a point (x0,y0) of radius δ  in the domain of the form

 (x-x0)2 + (y-y0)2 < δ2 .

Then the definition of limit takes on a strikingly similar form: The limit of f(x,y) as (x,y) approaches (x0,y0)  = L  means that

Given any ε there is a  δ such that  the value of z = f(x,y)  lies between the planes z = L - ε and

 z = L + ε

Whenever (x,y) is in the circle  (x-x0)2 + (y-y0)2 < δ2

 See Animation 1 -- Notice as we shrink the circle (let δ go to 0) centered at (x0,y0) the surface is confined between planes which are converging to z = 1.

Animation 1

Another way of thinking of limit is that if we approach (x0,y0) along any curve in the

domain we approach the same value on the surface. In the following animation we consider the

surface z = 1 - x2 - y2  (a parabaloid) , We consider what happens as we approach (0,0) along four

different lines. Note on the surface all values converge to 1. See Animation 2

Animation 2

Of course this is not a proof --we can't possibly consider the limit of every curve which approaches

 a point ( x0,y0). The formal  proof goes like this: We want to show z =1- x2 -y2 goes to 1 as (x,y) goes to (0,0)

Let ε be given then | f(x,y) - L | = | (1 - x2 - y2) -1| = |x2 + y2|

Let δ =√ ε    then if  x2 +  y2  <  δ2             | f(x,y) - L | < ε     

We can though use this to prove a limit does not exist as the next animation shows. If on the surface z approaches different values as we approach (x0,y0) along different paths then we say the limit does not exist.See animation 3. You'll notice that all 3 lines approach (0,0) in the domain but on the surface the corresponding z values approach 3 different values.                                                ,

Animation 3

Why doesn't this limit  exist ?

Here f(x,y) = (xy)/(x2 + y2)

suppose we approach (0,0) along any line of the form y = m x

f(x,mx) = mx2/[x2(m2+1)] = m/(m2+1)  hence the limit is m/(m2+1)  which can take on any value in  [-1/2,1/2]

The Smart Bunny-A very short story by Kurt Vonnegut Jr.

©2008-2010 Kelly Liakos