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.

> For most people—who won't solve complex math problems daily at work—the takeaway from learning math is not their mechanical ability at solving math problems. The takeaway is their understanding of math concepts and ideas, which will shape their thinking skills in general.

Agree, but for some others, there are real world consequences, e.g. whether they get accepted into a university or whether they can read and properly understand an academic paper.