
The sine function divided by its angle, and a certain limit
What is $ \lim_{x\rightarrow0} \frac{\sin x}{x}$ ? Recall the definition of a limit, repeated here for reference [2]. A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and only if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < xa < \delta $ we have $f(x) – A <…

The Radian
A radian is defined as follows. $360^{\circ} = 2 \pi$ radians. Dividing both sides by $2\pi$, one radian is: $ \frac{180}{\pi}^{\circ} = 1$ radian.

Exponents
The following rules apply for exponents that are positive integers. The justification is sketched here. These rules are also said to apply for exponents that are not positive integers–that is, zero and negative integers. If $ p$ and $ q$ are rational numbers, they are also said to apply. Finally, if $ p$ and…

Irrational Numbers
Are rational numbers the only numbers? The continuity of real numbers is an assumption in the CantorDedekind axiom [1]. The idea of continuity brings up the possibility of a different type of number called an irrational number that is, simply, not a rational number. So, if a number is not a rational number, it is…