# Delta Epsilon Proof

tylerc1991

## Homework Statement

Define: f(z) $$\rightarrow$$ w1 as z $$\rightarrow$$ z0
and
g(z) $$\rightarrow$$ w2 as z $$\rightarrow$$ z0

prove that f(z)/g(z) $$\rightarrow$$ w1/w2 as z$$\rightarrow$$ z0

## The Attempt at a Solution

let $$\epsilon$$ > 0
choose $$\delta$$ > 0 such that:
|f(z) - w1| < ______ (defined later)
|g(z) - w2| < ______ (defined later)

|f(z)/g(z) - w1/w2| = |f(z)/g(z) - f(z)/w2 + f(z)/w2 - w1/w2|
< |f(z)|*1/|g(z)-w2| + 1/|w2|*|f(z) -w1|

this is where I am stuck, I know that you have to make that add up to epsilon but I'm unsure how to pick them so it works out correctly. Any help would be greatly appreciated.

I'm not sure what you did to get to that second line from the first one, but it is surely not what you want because that 1/|g(z)-w2| becomes very large as z$\rightarrow$z0 since the denominator goes to 0. How about getting a common denominator first? Then this will look similar to the problem of proving that the product of the limits is the limit of the product.