It has been over 20 years since my last calculus class. Remember limits, derivatives, integrals and and calculating the area under a curve? If I remember correctly, limits were a particular pain, yet were fundamental to the understanding of calculus. Textbooks always gave definitions that were...well...too mathematical for my liking. Here's an example:
DEFINITION: (Graphical idea of a limit) A limit of a function f is the y-value approached by f as x approaches some fixed point. In particular:
(i) limx->a+ f(x) is the y-value that f approaches as x approaches a from the right.
(ii) limx->a− f(x) is the y-value that f approaches as x approaches a from the left.
(iii) If limx->a− f(x) = limx->a+ f(x), then we define limx->a f(x) to be their common value.
(iv) If limx->a− f(x) 6= limx->a+ f(x), then we say that limx->a f(x) does not exist.
Here is a better way to describe limits and it can be summed up in on very simple statement.
Eventually, it never gets there.