Mathematics 241 – Multivariable Calculus Quiz 3 – September 21, 2007

Your Name:

A) (5) Sketch the level curves with c = −1, 0, 1 for the function f(x, y) = x^{2} − y^{2}.

B) (5) Let

f(x, y) = \{ \begin{matrix}\frac{x+y}{\sqrt{x^{2}+y^{2}}} & if (x, y) ≠ (0, 0) \\ 0 & if (x, y) = (0, 0) \end{matrix}

Does f have a limit as (x, y) → (0, 0) along the line y = 0? (Be careful – +\sqrt{a^{2}} = |a|, not a.) What does your answer say about the existence of lim_{(x,y)→(0,0)} f(x, y)?