Is $L = \{(x,y,z) | x+y=z\}$ a regular language? Suppose $x,y,z$ are coded as decimal or their binary representations in an appropriate DFA. Is $L$ regular?
My intuition tells me that the answer is no, because there are infinitely many combinations such that $x+y=z$ and a DFA must contain a finite amount of memory. Is this correct?

No.  Say your numbers are written in base $b$, so your alphabet is symbols 0 to $b - 1$, and (,).
The proof is standard pumping lema:  suppose $L$ is regular, let $N$ be the pumping lema constant.  Take $\sigma =(10^N,10^N,20^N)$. The pumping lema tells you that a piece of the first number can be repeated without touching the other two. But doing so makes the equation fail, the resulting string isn't in the language. 
