Math 2060 - Section 14.2

2015 days ago by jimlb

Section 14.2

Worksheet by Jim Brown

In this section we explore limits of functions.  It can be helpful, especially when the limits do not exist, to see a graph of the function.  We give some examples in this worksheet.

Example 1: Consider the function $f(x,y) = \frac{y^4}{x^4 + 3y^4}$.  This is graphed here:

var('x,y'); P=plot3d(y^4/(x^4 + 3*y^4),(x,-1,1),(y,-1,1), aspect_ratio=true);P.show() 
       
(x, y)
(x, y)

One sees immediately from the graph that something funky is going on at the origin.  We would like to see if the limit as $(x,y) \rightarrow (0,0)$ exists.  We consider two paths.  Path one is graphed in red below and is given by $(x,0) \rightarrow (0,0)$.  The second path is graphed in yellow and is given by $(0,y) \rightarrow (0,0)$. 

P+=parametric_plot3d([x,0,0],(x,-1,1), color='red', thickness=3);P+=parametric_plot3d([0,y,1/3],(y,-1,1),color='yellow',thickness=5);P.show() 
       

It is clear that we do not have a limit at the origin; the red and yellow lines do not go to the same $z$ value!

Example 2:  In this example we consider the limit of $f(x,y) = \frac{3x^2 y}{x^2 + y^2}$ as $(x,y) \rightarrow (0,0)$.  We begin with a graph.

P= plot3d(3*x^2*y/(x^2+y^2), (x,-1,1),(y,-1,1),aspect_ratio=true);P.show() 
       

It looks like along each path we should get 0, so we prove this in the notes.

Example 3:  Let $f(x,y) = \frac{xy}{x^2 + y^2}$.  We want to determine if the limit of $f(x,y)$ as $(x,y) \rightarrow (0,0)$ exists.  We begin by graphing the function. 

P=plot3d(x*y/(x^2+y^2), (x,-1,1),(y,-1,1), aspect_ratio=true);P.show() 
       

This looks like a bit of a mess, so we try what we did in example 1.  Namely, consider the paths $(x,0) \rightarrow (0,0)$ and $(0,y) \rightarrow (0,0)$. These are graphed below (both in red), but note that $f(x,0) = 0$ and $f(0,y) = 0$, so these are both 0!

P+=parametric_plot3d([x,0,0], (x,-1,1), color='red', thickness=5);P+=parametric_plot3d([0,y,0], (y,-1,1), color='red', thickness=5);P.show() 
       

From the graph it seems the path towards $(0,0)$ where we take $x = y$ or $x=-y$ will give different values.  Try $x = y$.  Then $f(x,x) = \frac{x^2}{x^2 + x^2} = \frac{1}{2}$, so the limit is $\frac{1}{2}$.  Thus, the limit does not exist because we get the value of 0 along one path and $\frac{1}{2}$ along another path.  This path is graphed in yellow below.

P+=parametric_plot3d([x,x,1/2], (x,-1,1), color='yellow', thickness=5);P.show()