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8. The following problems are about limits and continuity.
(a) (4 pts)Show that
lim
(x,y)→(0,0)
2x2y
x2+y2= 0.
what can you say about the continuity of f(x, y) = 2x2y
x2+y2at the point (0,0)?
(b) (4 pts)Show that
lim
(x,y)→(0,0)
x2
x2+y2
does not exist. What can you say about the continuity of the function g(x, y) =
x2
x2+y2at the point (0,0)?
(c) (4 pts) Use level curves to show that
lim
(x,y)→(0,0)
xcos y+ysin y
x2+y2
does not exist. (You may use the applet in Mike May’s homepage, for which there
is a link on Lyceum, or any other graphing program you like).