Recall for the limits of functions of one variable

|f(x)-L| < e whenever |x-a| < d or a - d < x < a + d

what you are saying is when the distance from any point x to a < d

then |f(x)-L| < e

It's the same basic idea for functions of 2 variables except instead of an interval centered at a you use a circle centered at (a,b)

so whenever the distance from (a,b) to (x,y) < d then |f(x,y)-L| < e

using the distance formula this takes the form

if 0 < < δ then |f(x,y)-L| < e

Be Careful approaching (a,b) along a line segment y =cx never proves a limit but if you appraoach along 2 different lines, or curves for that matter, and get different results then the limit does not exist.

hope this helps a little