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:
(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 $(0,y) \rightarrow (0,0)$.
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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.
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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.
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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!
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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.
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