Part 1: Numerical Exploration of Limits

We will not study all possible functions of 2 variables, but will instead limit our attention to those functions of 2 variables which can be studied with the familiar concepts from calculus. To do so, however, requires that we develop a notion of limit for functions of 2 variables.

Intuitively, when we write

lim ( x,y) ® ( p,q)   f( x,y) = L

(1)

then we mean that if (x,y) approaches (p,q) , then f(x,y) approaches L. To illustrate this idea, let us define the Cartesian product [ a,b] ×[c,d] to be the rectangle in the xy-plane with one side corresponding to [ a,b] on the x-axis and the other corresponding to [ c,d] on the y-axis.

The interior of the rectangle [ a,b] ×[ c,d] is then defined to be the open rectangle ( a,b) ×( c,d) , which in set notation is defined

In addition, any open rectangle ( a,b) ×( c,d) which contains a point (p,q) is said to be a neighborhood of (p,q).

We then interpret (1) to mean that given any e > 0, there is an open rectangle ( a,b) ×(c,d) containing the point ( p,q) such that if (x,y) is in (a,b) ×(c,d) and (x,y) ¹ ( p,q) , then f( x,y) is within e of L.

That is, f( x,y) is between L-e and L+e, or equivalently, | f( x) -L| < e.      

Definition 2.1: We define the limit

lim ( x,y) ® ( p,q)   f( x,y) = L

(2)

to mean that for all e > 0, there is an open rectangle (a,b) × ( c,d) containing ( p,q) such that if ( x,y) is in ( a,b) ×(c,d) and if ( x,y) ¹ ( p,q) , then

| f( x,y) - L| < e

       

Moreover, definition 2.1 also implies that limits of 2 variables can be estimated by constructing a table in which the x's are approaching p, the y's are approaching q, and the interior entries represent f( x,y) evaluated at the implied points.       

EXAMPLE 1    Use a table to estimate the limit

lim ( x,y) ® ( 2,3)    x2y2-9x2-4y2+36 xy-3x-2y+6

Solution: To do so, we define

f( x,y) =  x2y2-9x2-4y2+36 xy-3x-2y+6

and then use f( x,y) to complete the table

x  \  y | 1.9 | f( 1.9,2.9) f( 1.9,2.99) 1.99 | f( 1.99,2.9) f( 1.99,2.99) 1.999 |  · · ·   · · · |  ???  2.001 |  · · ·    · ··  2.01 | 2.1 |

That is, we evaluate f( x,y) at all possible pairs to obtain

x  \  y | 2.9 2.99 2.999 3.001 3.01 3.1 1.9 | 23.01 23.361 5.999 23. 404 23.439 23.79 1.99 | 23.541 23.9 23.936 23. 944 23.98 24. 339 1.999 | 23.594 23.954 23.99 24.0 24.034 24.394 | ??? 2.001 | 23.606 23.966 24.0 24. 01 24.046 24.406 2.01 | 23.659 24.02 24.056 24.064 24.1 24.461 2.1 | 24.19 24.559 24.596 24.604 24. 641 25.01

The values in the vicinity of ??? appear to be approaching 24, so that we estimate that

lim ( x,y) ® ( 2,3)   x2y2-9x2-4y2+36 xy-3x-2y+6  = 24