- #1

looserlama

- 30

- 0

## Homework Statement

Let z = x + iy and let f(z) = 3xy + i(x - y

^{2}). Find lim

_{z→3 + 2i}f(z).

## Homework Equations

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 - y

^{2}) - 18 + i| ≤ |3xy - 18| + |1 + x - y

^{2}|

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 y

^{2}and the 3xy.

Any help would be awesome!