Give an example of the limit of a function at a boundary point of its domain?
(A) Domain
Let f be domain of function and f(x,y) be defined by
f(x,y)=(x2y22/(x2y2+(x−y)2)
The domain of this function is all values such that the denominator is not equal to 0.
Then we must simply find the solution to x2y2+(x−y)2=0 and this will be all points such that this is not true.
But notice xy=0 only if or for the real numbers. Then and is a solution.
Now if we observe all values (x−y)2 and x2y2as positive for all nonzero x and y, then we need only consider this point (0,0). Therefore the domain of f= R- (0,0)
(B) Boundary of domain
Let N be a neighborhood around (0,0) with radius ε>0
Then we have (ε-0)/2 is in N However, (0,0) is not in this set.
Since ε was arbitrary, we have that all neighborhoods of (0,0)contain points of the domain and points not in the domain.
Therefore (0,0) is a boundary point.
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