For me, the biggest stumbling block in understanding the usual ε/δ limit definition in high school was teachers reading |x - a| as "the absolute value of x minus a" rather than "the distance between x and a".
The later reading suggests a more intuitive (to me) definition: a limit f(x)→q as x→p exists if, for every open interval Y containing q, an open interval X containing p exists such that f(x)∊Y for all x≠p in X (and then if f(p)=q, f is also continuous at p).
Another nice property of the above definition: replace "interval" with "ball" or "neighborhood" for analogous definitions for functions between metric and topological spaces, respectively.
The later reading suggests a more intuitive (to me) definition: a limit f(x)→q as x→p exists if, for every open interval Y containing q, an open interval X containing p exists such that f(x)∊Y for all x≠p in X (and then if f(p)=q, f is also continuous at p).
Another nice property of the above definition: replace "interval" with "ball" or "neighborhood" for analogous definitions for functions between metric and topological spaces, respectively.