Exercise 17

Consider the function $h$ defined by $h(x,y) = \frac{2x^2y}{x^4 + y^2}.$

1. What is the domain of $h$?

2. Evaluate the limit of $h$ at $(0,0)$ along all linear paths the contain the origin. What does this tell us about $\lim_{(x,y) \to (0,0)} h(x,y)$?

3. Does $\lim_{(x,y) \to (0,0)} h(x,y)$ exist? Verify your answer. Check by using appropriate technology to sketch both surface and contour plots of $h$ near $(0,0)$. Write several sentences to say how your plots affirm your findings about $\lim_{(x,y) \to (0,0)} h(x,y)$.