Activity 11.2.2.
Consider the function \(f\text{,}\) defined by
\begin{equation*}
f(x,y) = \frac{y}{\sqrt{x^2+y^2}}
\end{equation*}
whose graph is shown below in Figure 11.2.16
(a)
Is \(f\) defined at the point \((0,0)\text{?}\) What, if anything, does this say about whether \(f\) has a limit at the point \((0,0)\text{?}\)
(b)
Values of \(f\) (to three decimal places) at several points close to \((0,0)\) are shown in Table 11.2.17.
| \(x\backslash y\) | \(-1.000\) | \(-0.100\) | \(0.000\) | \(0.100\) | \(1.000\) |
| \(-1.000\) | \(-0.707\) | — | \(0.000\) | — | \(0.707\) |
| \(-0.100\) | — | \(-0.707\) | \(0.000\) | \(0.707\) | — |
| \(0.000\) | \(-1.000\) | \(-1.000\) | — | \(1.000\) | \(1.000\) |
| \(0.100\) | — | \(-0.707\) | \(0.000\) | \(0.707\) | — |
| \(1.000\) | \(-0.707\) | — | \(0.000\) | — | \(0.707\) |
Based on these calculations, state whether you think \(f\) has a limit at \((0,0)\) and give an argument supporting your statement.
(c)
Now we formalize your conjecture from the previous part by considering what happens if we restrict our attention to different paths. First, we look at \(f\) for points in the domain along the \(x\)-axis; that is, we consider what happens when \(y = 0\text{.}\) Write out what the output of \(f\) will when \(y=0\text{.}\) In other words, \(f(x,0)= \text{?}\)
What is the behavior of \(f(x,0)\) as \(x \to 0\text{?}\) If we approach \((0,0)\) by moving along the \(x\)-axis, what value do we find as the limit?
(d)
What is the behavior of \(f\) along the line \(y=x\) when \(x \gt 0\text{;}\) that is, what is the value of \(f(x,x)\) when \(x>0\text{?}\) If we approach \((0,0)\) by moving along the line \(y=x\) in the first quadrant (thus considering \(f(x,x)\) as \(x \to 0^+\)), what value do we find as the limit?
(e)
In general, if \(\displaystyle{\lim_{(x,y)\to(0,0)}f(x,y) = L}\text{,}\) then \(f(x,y)\) approaches \(L\) as \((x,y)\) approaches \((0,0)\text{,}\) regardless of the path we take in letting \((x,y) \to (0,0)\text{.}\) Based on the last two parts of this activity, can the limit
\begin{equation*}
\lim_{(x,y)\to(0,0)} f(x,y)
\end{equation*}
exist? Write a sentence or two to justify your idea.
(f)
Substitute \(y=mx\) into \(f(x,y)\) and simplify your result for \(f(x,mx)\text{.}\) Use your expression and limit rules to evaluate
\begin{equation*}
\lim_{x\rightarrow 0} f(x,mx)
\end{equation*}
Use this limit to explain the the behavior of \(f(x,y)\) as \((x,y)\) approaches \((0,0)\) along line of the form \(y=mx\) and how this observation reinforces your conclusion about the existence of \(\displaystyle{\lim_{(x,y)\to(0,0)}f(x,y)}\) from the previous part of this activity?
