Weierstrass' method for handling limits with the epsilon and delta symbols is very useful for rigorous analysis of math but it is terrible in terms of any intuitive approach to limits. There are are other ways to teach limits. For instance with infinitesimals yet I don't know if that has been simplified. Why does the 'epsilon- delta' method seem to dominate many calculus courses? Are there Calculus Courses that don't use this 'epsilon-delta' method?
Teaching calculus using infinitesimals from nonstandard analysis has been tried, Keisler's text is considered a staple in this regard. But I am curious as to what is so "terrible" about epsilon-delta approach? Using infinitesimals invites us to envision a hidden world of numbers beyond the real numbers, into which all elementary functions miraculously extend, and their properties there are then mysteriously reflected back into the world of real numbers.
The intuitive idea of a limit, as taught to non-majors, is "$f(x)$ approaches $L$ as $x$ approaches $a$", epsilon and delta simply quantify what "approaches" means for $f(x)$ and $x$ respectively. Admittedly, epsilon and delta arguments are technical, but it does not mean that they are unintuitive. We know from the early history of calculus that the simplicity of intuitions about limits is misleading because those intuitions are self-contradictory. Resolving these contradictions inevitably leads to subtleties and technicalities, and refinement of intuition, under any systematic approach. At least, epsilon and delta does it without invoking extra phantom entities, that would have to be either explained or glossed over in addition to real numbers, so why is it worse than infinitesimals?
This is not to say that calculus with epsilon and delta can not be taught terribly, and that unfortunately it often is. But I am not sure that this is the fault of the approach rather than of the fact that there is a fundamental difficulty in the concept itself, as its 2000 year long genesis suggests. Instructors are often unable and/or unwilling to dedicate the time and the effort required to teach it properly. Especially since the algorithmic aspect of calculus is so much easier to teach, and can be taught while ignoring limits almost completely.
The idea that historical infinitesimals were self-contradictory is prevalent among historians (see e.g., the accepted answer above) and also many mathematicians, but it has been challenged in the recent literature; see e.g., this article in Erkenntnis as well as its MathSciNet review. Furthermore this recent study presents the results of a questionnaire that indicate that students themselves overwhelmingly prefer infinitesimal definitions a la Cauchy of basic calculus concepts like continuity and derivative. The reason epsilon-delta paraphrases still dominate in teaching are due to a history of prejudice.