Activity 11.3.3

Define the function $g$ by $$\begin{gathered} g(x,y) = \frac{x^2y}{x^4 + y^2} \end{gathered}$$. We will investigate the limit $\displaystyle{\lim_{(x,y)\to(0,0)}g(x,y)}$.

1. 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?

2. 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?

3. What is the behavior of $g$ on the line $y=mx$? That is, what is $g(x,mx)$ and what is the limit of $g$ as $(x,y)$ approaches $(0,0)$ along the line $y=mx$?

4. Based on what you have seen so far, do you think $\displaystyle\lim_{(x,y)\to(0,0)}g(x,y)$ exists? If so, what do you think its value is?

5. Now consider the behavior of $g$ on the parabola $y=x^2$? What is $g(x,x^2)$ and what is the limit of $g$ as $(x,y)$ approaches $(0,0)$ along this parabola?

6. 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.