Activity 11.2.2.
Consider the function \(f\text{,}\) defined by
\begin{equation*}
f(x,y) = \frac{y}{\sqrt{x^2+y^2}}\text{.}
\end{equation*}
The graph of this function is shown below.
(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)
| \(x\backslash y\) | \(-1.000\) | \(-0.100\) | \(0.000\) | \(0.100\) | \(1.000\) |
| \(-1.000\) | \(-0.707\) | — | \(0.000\) | — | \(0.707\) |
| \(-0.100\) | — | \(-0.707\) | \(0.000\) | \(0.707\) | — |
| \(0.000\) | \(-1.000\) | \(-1.000\) | — | \(1.000\) | \(1.000\) |
| \(0.100\) | — | \(-0.707\) | \(0.000\) | \(0.707\) | — |
| \(1.000\) | \(-0.707\) | — | \(0.000\) | — | \(0.707\) |
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, what is \(f(x,0)\) equal to?
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, 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 explain your reasoning.
(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)\text{.}
\end{equation*}
Use this limit to explain the the behavior of \(f(x,y)\) as \((x,y)\) approaches \((0,0)\) along lines 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.
