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

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

Solution: The level curve for the value c is x^{2} − y^{2} = c. When c = 0, we get the two lines y = ±x. With c ≠ 0, the level curve is a hyperbola. The hyperbola opens left and right if c > 0. The hyperbola opens up and down if c < 0. Here is a plot, from Maple:

B) (5) Let

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

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)?

Solution. If y = 0, then f(x, 0) = \frac{x}{\sqrt{x^{2}}} = \frac{x}{|x|}, which is +1 if x > 0 and −1 if x < 0. This says,

lim_{x→0^{−}} f(x, 0) = −1 ≠ 1 = lim_{x→0^{+}} f(x, 0).

Since there is not even a single limit along this one line, the limit of f(x, y) as (x, y) → (0, 0) cannot exist.