26. Ev In Exer (a) @(x, y) is continuous for all (x, y). 27. (b) @(x, y) is continuous for (x, y) # (0, 0). (2,2 (c) @(x, y) # 0 for all (x, y). 29. (x , ] 4. Suppose that f(x, 0) = 3 for all x # 0 and f (0, y) = 5 for all y # 0. What can you conclude about lim f(x, y)? (x, y) - (0,0) 31. (x , 33. (x, 14. Let f(x, y) = xy/(x2 + yz). Show that f(x, y) approaches zero along 35. the x- and y-axes. Then prove that lim f(x, y) does not exist by (x , (x, y) -(0,0) showing that the limit along the line y = x is nonzero. 37. (x , 15. Let f(x, y) = *5 + y3 xy 2 -. Set y = mx and show that the resulting limit 39. depends on m, and therefore the limit lim f(x, y) does not exist. (x , (x, y) - (0,0) 16. Let f(x, y) = 2x2 + 3 y2 Set y = mx and show that the resulting xy Furth limit depends on m, and therefore the limit lim f(x, y) does not (x, y) - (0;0) 46. Ev exist. 47. Is 17. Prove that 19 lim (x, y) -(0,0) x2+ 2 does not exist by considering the limit along the x-axis. 18. Let f(x, y) = x3 / (x2 + y2) and g(x, y) = x2/(x2 + 2). Using polar 48. C coordinates, prove that xy 0 (a) Is lim is a con itllamaand (onit gable (x,y) (0,0) f ( x , y ) = 0 (b) Us conclus and that lim (x,y)-(0,0) 8 (4, y) does not exist. Hint: Show that g(x, y) = coste and observe that cose can take on any value between -1 and 1 as avon (x, y) - (0, 0)