In questions 17-20, determine whether the statement is true or false. Explain your answer.
17. If the domain of f(x, y) is the xy-plane, then the domain of f(sin^{-1} t, sqrt{t}) is the interval [0,1].
18. If f(x, y) = y/x, then a contour f(x, y) = m is the straight line y = mx.
19. The natural domain of f(x, y, z) = sqrt{1 - x^2 - y^2} is a disk of radius 1 centered at the origin in the xy-plane.
20. Every level surface of f(x, y, z) = x + 2y + 3z is a plane.
In questions 21 & 22, show that the limit does not exist by considering the limits as (x, y) → (0, 0) along the coordinate axes.
21. (a) lim_{(x,y)→(0,0)} 3 / (x^2 + 2y^2)
(b) lim_{(x,y)→(0,0)} (x + y) / (2x^2 + y^2)
22. (a) lim_{(x,y)→(0,0)} (x - y) / (x^2 + y^2)
(b) lim_{(x,y)→(0,0)} (cos xy) / (x^2 + y^2)
In questions 23-28 determine whether the limit exists. If so, find its value.
23. lim_{(x,y)→(0,0)} (x^4 - y^4) / (x^2 + y^2)
24. lim_{(x,y)→(0,0)} (xy) / (3x^2 + 2y^2)
25. lim_{(x,y)→(0,0)} (x^4 - 16y^4) / (x^2 + 4y^2)
26. lim_{(x,y)→(0,0)} (1 - x^2 - y^2) / (x^2 + y^2)
27. lim_{(x,y,z)→(2,-1,2)} (xz^2) / sqrt{x^2 + y^2 + z^2}
28. lim_{(x,y,z)→(2,0,-1)} ln(2x + y - z)
In questions 29 & 30, determine whether the statement is true or false. Explain your answer.
29. If D is an open set in 2-space or in 3-space, then every point in D is an interior point of D.
30. If f(x, y) → L as (x, y) approaches (0, 0) along the x-axis, and if f(x, y) → L as (x, y) approaches (0, 0) along the y-axis, then lim_{(x,y)→(0,0)} f(x, y) = L.