Let z = x + iy and let f(z) = 3xy + i(x - y2). Find limz→3 + 2i f(z).
The definition of a limit.
The Attempt at a Solution
I did f(3 + 2i) = 18 - i
It seems pretty clear that it is a continuous function, but I can't prove it.
So I tried using the definition of a limit of a function:
Let ε be a strictly positive real number.
Take |x + iy - 3 -2i| ≤ |x - 3| + |y - 2| < δ
Then, |3xy + i(x - y2) - 18 + i| ≤ |3xy - 18| + |1 + x - y2|
Clearly I need to get to some multiple of |x - 3| + |y - 2| which would give me a relation between δ and ε.
The problem is I can get there. I tried adding and subtracting some terms in the absolute values, but I have trouble getting rid of the y2 and the 3xy.
Any help would be awesome!