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Activity 11.2.3 .
Let’s consider the function \(g\) defined by
\begin{equation*}
g(x,y) = \frac{x^2y}{x^4 + y^2}
\end{equation*}
and investigate the limit \(\displaystyle{\lim_{(x,y)\to(0,0)}g(x,y)}\text{.}\)
(a) What is the behavior of
\(g\) on the
\(x\) -axis? That is, what is
\(g(x,0)\) and what is the limit of
\(g\) as
\((x,y)\) approaches
\((0,0)\) along the
\(x\) -axis?
(b) What is the behavior of
\(g\) on the
\(y\) -axis? That is, what is
\(g(0,y)\) and what is the limit of
\(g\) as
\((x,y)\) approaches
\((0,0)\) along the
\(y\) -axis?
(c) What is the behavior of
\(g\) on the line
\(y=mx\text{?}\) That is, what is
\(g(x,mx)\) and what is the limit of
\(g\) as
\((x,y)\) approaches
\((0,0)\) along the line
\(y=mx\text{?}\)
(d) Based on what you have seen so far, do you think
\(\lim_{(x,y)\to(0,0)}g(x,y)\) exists? If so, what do you think its value is?
(e) Now consider the behavior of
\(g\) on the parabola
\(y=x^2\text{?}\) What is
\(g(x,x^2)\) and what is the limit of
\(g\) as
\((x,y)\) approaches
\((0,0)\) along this parabola?
(f) State whether you think the limit
\(\displaystyle{\lim_{(x,y)\to(0,0)} g(x,y)}\) exists or not and provide a justification of your statement.
