Show that lim (x, y)→(0, 0) 2x2 − 5y2 x2 + y2 does not exist. SOLUTION Let f(x, y) = (2x2 − 5y2)/(x2 + y2). First, let's approach (0, 0) along the x-axis. Then y = gives f(x, 0) = 2x2/x2 = for all x ≠ 0, so f(x, y) → as (x, y) → (0, 0) along the x-axis. We now approach along the y-axis by putting x = . Then f(0, y) = −5y2/y2 = for all y ≠ 0, so f(x, y) → as (x, y) → (0, 0) along the y-axis. Since f has two different limits along two different lines, the given limit does not