Exercise 15

Consider the function $f$ defined by $f(x,y) = \frac{xy}{x^2 + y^2 + 1}.$

1. What is the domain of $f$?

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

3. What do you conjecture is the value of $\lim_{(x,y) \to (0,0)} f(x,y)$?

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

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