Exercise 16

Consider the function $g$ defined by $g(x,y) = \frac{xy}{x^2 + y^2}.$

1. What is the domain of $g$?

2. Evaluate limit of $g$ at $(0,0)$ along the following paths: $x = 0$, $y = x$, and $y = 2x$.

3. What can you now say about the value of $\lim_{(x,y) \to (0,0)} g(x,y)$?

4. Is $g$ continuous at $(0,0)$? Why or why not?

5. Use appropriate technology to sketch both surface and contour plots of $g$ near $(0,0)$. Write several sentences to say how your plots affirm your findings in (a) - (d).