Page 11 - Calc for the Clueless

We use the identity tan x = sin x/cos
x.

Therefore

How are you doing so far? Let's put in one more example.

Example 17—

Note

2
In Example 3, if sin x were in the bottom and x were in the top, then the limit would be 1/0, which would be
undefined.

Formal Definition

We will now tackle the most difficult part of basic calculus, the theoretical definition of limit. As previously
mentioned, it took two of the finest mathematicians of all times, Newton and Leibniz, to first formalize this
topic. It is not essential to the rest of basic calculus to understand this definition. It is hoped this explanation
will give you some understanding of how really amazing calculus is and how brilliant Newton and Leibniz must
have been. Remember this is an approximating process that many times gives exact (or if not, very, very close)
answers. To me this is mind-boggling, terrific, stupendous, unbelievable, awesome, cool, and every other great
word you can think of.

Definition

if and only if, given ε > 0, there exists δ > 0 such that if 0 < |x - a | < δ, then |f(x) - L| < ε.

Note

ε = epsilon and δ = delta—two letters of the Greek alphabet.

Translation 1

Given ε, a small positive number, we can always find δ, another small positive number, such that if x is within a
distance δ from a but not exactly at a, then f(x) is within a distance ε from L.