Vectors Calaulus 1.11

Limits Topology Terminology Let P be a point in the plane then a d-neighborhood (ball) of P is the set of points that are less than d units away from P.  If R is a region, then a point P is called an interior point  of R if there is a d-neighborhood totally contained in R.  If every point of R is an interior point of R then R is called open. A point P is called a boundary point of R if every d-neighborhood  of R contains both points in R and not in R.  R is called closed if it contains all of its boundary points.            The Definition of a Limit                           Definition Let f(x,y) be a function defined near the point P, then         if there is a d such that f(x,y) is close to L for all points (except possibly P) in the d - neighborhood of P. Equivalently, the limit is L if for all paths that lead to P, the function also tends towards P.   (Recall that for the one variable case we needed to check only the path from the left and  from the right.) To show that a limit does not exist at a point we need only find two paths that both lead to P such that f(x,y) tends towards different values. Techniques For Finding Limits Example Show that               Does not exist   Solution  First select the path along the x-axis.  On this path         y = 0 so the function becomes:                           0         f(x,0)  =              =  0                          x2   Now choose the path along the y = x line:                           x2            1         f(0,y)  =              =                                    2x2           2   Hence the function tends towards two different values for different paths.  We can conclude that the limit does not exist.  The graph is pictured below. Example Find         We could try the paths from the last example, but both paths give a value of 0 for the limit.   Hence we suspect that the limit exists.  We convert to polar coordinates and take the limit  as r approaches 0: We have                           r3cos3q + r3sin3q         f(r,q)  =                                    =  rcos3q + rsin3q                                     r2   as r approaches 0 the function also approaches 0 no matter what q is.  Hence the limit is 0. Below is the graph of this function. Exercises:  Find the limit if it exists Continuity We make the following definition for continuity.                               Definition A function of several variables is continuous at a point P if the limit exists at P and the function defined at P is equal to this limit. As with functions of one variable, polynomials are continuous, sums, products, and  compositions of continuous functions are continuous.  Quotients of continuous functions  are continuous.  A function is continuous if it is continuous at every point.