EXAMPLE 6 Functions of three variables x² sin y lim (x, y, 2)→(2, /2, 0) z² + 4 a. Evaluate b. Find the points at which h(x, y, z) = Vx² + y² + z?• 1 is continuous.

Suppose f is a differentiable function of x and y, and g(r, s) = f(5r – s, s² - 6r). Use the table of values below to calculate g,(6, 3) and g,(6, 3). (27, -27) 4 8 1 (6, 3) 7 5 9 g,(6, 3) = g,(6, 3) = Additional Materials

2. Is it possible for a function f(x, y) to satisfy both ƒ(1, 2) = 5 and f(0,0) = 5? If yes, give an example; if no, explain why. (Hint: Two different inputs, but the same output?)

Please explain each property (withs #'s) V Consider the function d: R² x R² → R defined by d(x, y): min{|y2 - x₂, 1} if x₁ = y₁ otherwise, for x = (x₁, x₂), y = (₁, 2) (That is, for example for x = (1,2) and y = (2,3), d(x, y) = 1, for x = (1, 2) and y = (1, 1.5), d(x, y) = 0.5, and for x = (1,2), y = (1, 3), d(x, y) = 1.) Is this a well-define metric on R²? Argue for each condition of a metric if it is satisfied and provide a justification or counterexample.