Limits in the Plane

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(5x))/x),defined for x≠0.

Let's consider the limit as x goes to 0 graphically.

Plot[((Sin[5x])/x),{x,-.1,.1}]

From the plot, we can see that if epsilon=0.2, then we can choose delta=0.09 (the function values are between 4.80 and 5.00 for  0<|x|<0.09).
Suppose we want a smaller value of epsilon, like 0.01. We can readjust delta (or d in the command below), until our function values are between 4.99 and 5.00.

d:=0.02;Plot[((Sin[5x])/x),{x,-d,d}]

Now let’s consider the function f(x)=(x/(|x|)),defined for x≠0.

d:=0.02;Plot[(x/(Abs[x])),{x,-d,d}]

As we adjust the value of delta (d), the function values stay between 1 and -1. So if epsilon is 0.2, we cannot find a delta that forces our function values to stay within a range of 0.2 from any number. The limit as x goes to 0 does not exist in this case.



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

f:=(((x^2)-(y^2))/((x^2)+(y^2)))

d:=1;Plot3D[f,{x,-d,d},{y,-d,d}]

Now consider the function g(x,y)=(xy/((x^2)+(y^2))), defined for (x,y)≠(0,0).

g:=((x*y)/((x^2)+(y^2)))

d:=1;Plot3D[g,{x,-d,d},{y,-d,d}]

Next, consider the function h(x,y)=((xy^2)/((x^2)+(y^4))),defined for (x,y)≠(0,0).

h:=((x*(y^2))/((x^2)+(y^4)))

d:=1;Plot3D[h,{x,-d,d},{y,-d,d}]

Finally, we consider the function k(x,y)=((xy^2)/((x^2)+(y^2))),defined for (x,y)≠(0,0).

k:=((x*(y^2))/((x^2)+(y^2)))

d:=1;Plot3D[k,{x,-d,d},{y,-d,d}]