Multi-Variable Calculus

Definition of a Function of Two Variables

Limits

Limits for Functions of One Variable. What do we mean when we say that Informally, we might say that as x gets “ closer and closer ” to a, f(x) should get “ closer and closer ” to L.

Recall the definition of Limit: In less formal language this means that, if the limit holds, then f(x) gets closer and closer to L as x gets closer and closer to c. c ( ) L x is the input f(x) is the output

Defining the Limit a L Means that given any tolerance T for L we can find a tolerance t for a such that if x is between a-t and a+t, but x is not a, f(x) will be between L-T and L+T. L+T L-T a+ta-t (Graphically, this means that the part of the graph that lies in the black vertical strip---that is, those values that come from (a-t,a+t)--- will also lie in the blue horizontal strip.) Remember: the pt. (a,f(a)) is excluded!

a L L+T L-T No amount of making the Tolerance around a smaller is going to force the graph of that part of the function within the blue strip! This isn ’ t True for This function!

Changing the value of L doesn ’ t help either! a L L+T L-T

Functions of Two Variables How does this extend to functions of two variables? We can start with informal language as before: means that as (x,y) gets “ closer and closer ” to (a,b), f(x,y) gets closer and closer to L.

“ Closer and Closer ” The words “ closer and closer ” obviously have to do with measuring distance. In the real numbers, one number is “ close ” to another if it is within a certain tolerance---say no bigger than a+.01 and no smaller than a-.01. In the plane, one point is “ close ” to another if it is within a certain fixed distance---a radius! (a,b) r

What about those strips? (a,b) r (x,y) The vertical strip becomes a cylinder!

Horizontal Strip? L L+T L-T The horizontal strip becomes a “ sandwich ” ! Remember that the function values are back in the real numbers, so “ closeness ” is once again measured in terms of “ tolerance. ” The set of all z-values that lie between L-T and L+T, are “ trapped ” between the two horizontal planes z=L-T and z=L+T L lies on the z-axis. We are interested in function values that lie between z=L-T and z=L+T

Putting it All Together The part of the graph that lies above the green circle must also lie between the two horizontal planes. if given any pair of horizontal planes about L, we can find a circle centered at (a,b) so that the part of the graph of f within the cylinder is also between the planes.

Defining the Limit Means that given any tolerance T for L we can find a radius r about (a,b) such that if (x,y) lies within a distance r from (a,b), with (x,y) different from (a,b), f(x,y) will be between L-T and L+T. L+T L-T L

Definition of the Limit of a Function of Two Variables

x y z Example : suppose that If the limit holds, we should be able to construct a circle centered at (-1,3) with as the radius and any point inside this circle will generate a z value that is closer to 13 than.25. Center (-1,3) (x,y)

Functions of Two Variables How does this extend to functions of two variables? We can start with informal language as before: means that as (x,y) gets “ closer and closer ” to (a,b), f(x,y) gets closer and closer to L.

“ Closer and Closer ” The words “ closer and closer ” obviously have to do with measuring distance. In the real numbers, one number is “ close ” to another if it is within a certain tolerance---say no bigger than a+.01 and no smaller than a-.01. In the plane, one point is “ close ” to another if it is within a certain fixed distance---a radius! (a,b) r

What about those strips? (a,b) r (x,y) The vertical strip becomes a cylinder!

Horizontal Strip? L L+T L-T The horizontal strip becomes a “ sandwich ” ! Remember that the function values are back in the real numbers, so “ closeness ” is once again measured in terms of “ tolerance. ” The set of all z-values that lie between L-T and L+T, are “ trapped ” between the two horizontal planes z=L-T and z=L+T L lies on the z-axis. We are interested in function values that lie between z=L-T and z=L+T

Definition of the Limit of a Function of Two Variables

Our definition says that the distance between f(x, y) and L can be made arbitrarily small by making the distance from (x, y) to (a, b) sufficiently small (but not 0). The definition refers only to the distance between (x, y) and (a, b). It does not refer to the direction of approach.

Our definition says that the distance between f(x, y) and L can be made arbitrarily small by making the distance from (x, y) to (a, b) sufficiently small (but not 0). The definition refers only to the distance between (x, y) and (a, b). It does not refer to the direction of approach.

Therefore, if the limit exists, then f(x, y) must approach the same limit no matter how (x, y) approaches (a, b). Thus, if we can find two different paths of approach along which the function has different limits, then f(x, y) has no limit as (x, y) approaches (a, b).

Example-1

Example-2 If f(x, y) = xy/(x 2 + y 2 ), does exist?

work in pairs

Limit Laws The Limit Laws listed in one variable function can be extended to functions of two variables: The limit of a sum is the sum of the limits, The limit of a product is the product of the limits, and so on. The Squeezing Theorem also holds.

Example:

Try Me :

How about Me?

Class/Home work Page-682 Section 14.2 Exercises 3, 4, 8, 11, 16, 23, 26,