By considering different paths of approach, show that the function below has no limit as (x,y)→(0,0). f(x,y) = %3D .4 .2 + Examine the values of f along curves that end at (0,0). Along which set of curves is fa constant value? O A. y= kx, x#0 O B. y= kx + kx², x#0 O C. y= kx°, x#0 O D. y= kx?, x+0 If (x,y) approaches (0,0) along the curve when k = 1 used in the set of curves found above, what is the limit? (Simplify your answer.) If (x,y) approaches (0,0) along the curve when k = 0 used in the set of curves found above, what is the limit? (Simplify your answer.) What can you conclude? A. Since f has two different limits along two different paths to (0,0), in cannot be determined whether or not f has a limit as (x,y) approaches (0,0). B. Since f has two different limits along two different paths to (0,0), by the two-path test, f has no limit as (x,y) approaches (0,0). O C. Since f has the same limit along two different paths to (0,0), by the two-path test, f has no limit as (x,y) approaches (0,0).

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