Page 12 - Calc for the Clueless

Translation 2

We will explain this definition using an incorrect picture. I feel this gives you a much better idea than the
correct picture, which we will use next.

Interpret |x - a| as the distance between x and a, but instead of the one-dimensional picture it really is, imagine
that there is a circle around the point a of radius δ. |x- a| < δ stands for all x values that are inside this circle.
Similarly, imagine a circle of radius ε around L, with |f(x) - L| < ε the set of all points f(x) that are inside this
circle.

The definition says given ε > 0 (given a circle of radius ε around L), we can find δ > 0 (circle of radius δ around
a) such that if 0 < |x - a| < δ (if we take any x inside this circle), then |f(x) - L| < ε (f(x)) will be inside of the
circle of radius ε but not exactly at L.

Now take another ε ε 2, positive but smaller than ε (a smaller circle around L); there exists another δ δ 2, usually a
smaller circle around a, such that if 0 < |x - a| < δ 2, then |f(x) - L| < ε 2.

Now take smaller and smaller positive ε's; we can find smaller and smaller δ's. In the limit as the x circle
shrinks to a, the f(x) circle shrinks to L. Read this a number of times.!!!

Translation 3

Let us see the real picture. y = f(x). |x - a| < δ means a - δ < x < a+δ. |y-L| < ε means L - ε < y < L + ε.