Limits in the Plane
Math 211  Prof. Leise

Email completed labs to tleise@amherst.edu by noon next Wednesday.

Let's begin with a limit of a single variable function f(x) to review the concept of limit and visualize the roles of epsilon and delta.

The limit of f(x) as x approaches a is L if for every epsilon ϵ>0 we can find delta δ>0 such that |f(x)-L|<ϵ whenever 0<|x-a|<δ.

Consider the function f(x)=((sin(4x))/(2x)),defined for x≠0.  We can use the limit command in Mathematica to determine the limit as x approaches 0:

Limit[((Sin[4x])/(2x)),x->0]

Let's consider the limit graphically.

Plot[((Sin[4x])/(2x)),{x,-.1,.1}]

From the plot, we can see that if epsilon=0.05, then we can choose delta=0.09 (the function values are between 1.95 and 2.00 for  0<|x|<0.09).

Exercise 1. For the function f(x)=((sin(4x))/(2x))and epsilon=0.01, find a value for delta such that |f(x)-2|<0.01 whenever 0<|x-0|<δ.  You may do this by plotting the function for smaller intervals of x until the range is within 0.01 of 2.

Exercise 2. For the function f(x)=((sin(4x))/(2x))and epsilon=0.001, find a value for delta such that |f(x)-2|<0.001 whenever 0<|x-0|<δ.  You may do this by plotting the function for smaller intervals of x until the range is within 0.001 of 2.

Now let's move on to limits in the plane.  Consider the function f(x,y)=(xy/((x^2)+xy+(y^2))), defined for (x,y)≠(0,0).

Exercise 3. What is the limit of f(x,y) approaching (0,0) along the x-axis?  Along the y-axis?  Along the line y=x?  Does the limit exist?  (You can just do this in your head or on paper, and type your answer.)

Exercise 4. Plot f(x,y) using the Plot3D command for -1<x<1 and -1<y<1.

Exercise 5. Plot f(x,y) using the Plot3D command for -0.01<x<0.01 and -0.01<y<0.01 and then an even smaller domain like  -0.001<x<0.001 and -0.001<y<0.001.  Is the range of f(x,y) shrinking as the plotting domain shrinks?  (Compare the values of the z-axis in Exercises 4 and 5.)

Exercise 6. Based on these plotting results, is it possible to find a delta for every epsilon in the limit definition as (x,y) approaches (0,0)?  Is this consistent with your answer to Exercise 3?

Now consider the function f(x,y)=(((x+1)(y^2)+(x^2)(y+1))/((x^2)+(y^2))).

Exercise 7. Use polar coordinates to find the limit of f(x,y) as (x,y) approaches (0,0).  (You may do this on paper or in your head or by using Mathematica after substituting in polar coordinates).

Exercise 8. Plot f(x,y) using the Plot3D command for -1<x<1 and -1<y<1.

Exercise 9. Plot f(x,y) using the Plot3D command for -0.01<x<0.01 and -0.01<y<0.01and then an even smaller domain like  -0.001<x<0.001 and -0.001<y<0.001.  Is the range of f(x,y) shrinking as the plotting domain shrinks?  Does the limit of f(x,y) as (x,y) approaches (0,0) exist?

Exercise 10. For the function  f(x,y)=(((x+1)(y^2)+(x^2)(y+1))/((x^2)+(y^2)))and epsilon=0.001, find a value for delta such that |f(x,y)-1|<0.001 whenever 0<(Sqrt[(x^2)+(y^2)])<δ.  You may do this by plotting the function for smaller squares until the range is within 0.001 of 1 (from the graph for Exercise 9, you should be able to see that delta=0.01 works for epsilon=0.01).  Note that the plot is over a square that contains the disc we are interested in; if delta works for the square [-delta,delta]x[-delta,delta], then it works for the disc of radius delta.