How can I correctly justify, demonstrate, and properly determine that this limit exists?
lim (x,y) --> (0,0) $\displaystyle \sqrt{x^2+y^2}$
To prove this by the definition, you should state that
"For all x,y in R, for all epsilon > 0, there exists a delta > 0 such that [d(x, y) < delta] implies [sqrt(x^2 + y^2) < epsilon]
since "d(x, y)" and "sqrt(x^2 + y^2)" are the same thing, you can choose delta to be the epsilon that is given.