E finally consider f along rays emanating from the

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(e) Finally, consider f along rays emanating from the origin. Note that these are given by y = mx , for some (constant) value of m . Find and simplify f on the ray y = x : f ( x , x ) = (Notice that this means that y = x is a contour of f. Be sure you can explain why this is.) Find and simplify f on any ray y = mx . f ( x , mx ) = (Again, notice that this means that any ray y = mx is a con- tour of f; be sure you can explain why.) (f) Is f ( x , y ) continuous at ( 0 , 0 ) ? ? Correct Answers: 0 0 0 yes 0 0 0 yes 5/2 5*m/(mˆ2 + 1) no 23. (1 pt) Show that the function f ( x , y ) = x 5 y x 10 + y 5 . does not have a limit at ( 0 , 0 ) by examining the following limits. (a) Find the limit of f as ( x , y ) ( 0 , 0 ) along the line y = x . lim ( x , y ) ( 0 , 0 ) y = x f ( x , y ) = (b) Find the limit of f as ( x , y ) ( 0 , 0 ) along the line y = x 5 . lim ( x , y ) ( 0 , 0 ) y = x 5 f ( x , y ) = (Be sure that you are able to explain why the results in (a) and (b) indicate that f does not have a limit at (0,0)! Correct Answers: 0 1 24. (1 pt) Find the limit (enter ’DNE’ if the limit does not exist) lim ( x , y ) ( 0 , 0 ) ( 2 x + y ) 2 4 x 2 + y 2 1) Along the x-axis: 2) Along the y-axis: 3) Along the line y = x : 4) The limit is: Correct Answers: 1 1 1.8 4
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DNE 25. (1 pt) The largest set on which the function f ( x , y ) = 1 / ( 6 - x 2 - y 2 ) is continuous is A. The exterior of the circle x 2 + y 2 = 6 B. All of the xy-plane except the circle x 2 + y 2 = 6 C. All of the xy-plane D. The interior of the circle x 2 + y 2 = 6 E. The interior of the circle x 2 + y 2 = 6, plus the circle Correct Answers: B Generated by c WeBWorK, , Mathematical Association of America 5
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