Problem Set 11, Due Thirteenth Class Meeting
(1) Use the definition of limit to show that
f(x, y) =
0 if (x, y) = (0,0)
x2y3
x2+y2if (x, y)6= (0,0)
converges to 0 as (x, y)→(0,0). Is fcontinuous at (0,0)? Explain.
(2) Use the definition of limit to show that
f(x, y) =
0 if (x, y) = (0,0)
x2
x2+y2if (x, y)6= (0,0)
does not converges to 0 as (x, y)→(0,0). Is fcontinuous at (0,0)? Explain.
(3) Use the definition of limit to show that if a > 0 and b > 0 then
f(x, y) =
0 if (x, y) = (0,0)
|x|a|y|b
x2+y2if (x, y)6= (0,0)
converges to 0 as (x, y)→(0,0) if a+b > 2 and is not continuous at (0,0)
if a+b≤2.
1
