Why Read This Book? - Index of

7.2 Limits and Their Basic Properties 215
a = 0. Except for a passing glance at sine in Section 6.1, we have never mentioned
any trigonometric, exponential, or logarithmic functions. There is a reason for this:
Definitions of these functions that are rooted in the axioms of the real numbers
are not possible to come by at this stage of our game. Only the algebraic functions
arise from the theory we have developed thus far, and they can make for
some pretty sticky proofs by themselves. Strict definitions of these nonalgebraic
functions, called transcendental functions, come later. However, if we kick back
for a while and give ourselves the freedom to talk about sine, cosine, and tangent
in the familiar language of the unit circle, then we can apply Exercise 7.2.16
to show
sin x
lim = 1 (7.5)
x→0 x
In trigonometry, we take a real x-number line and “wrap” it around the unit
circle in the uv-plane with x = 0 placed at (u, v) = (1, 0) and the positive half
of the x-axis wrapped counterclockwise. (See Fig. 7.3.) For a real number x, we
define cos x and sin x to be the u and v coordinates, respectively, of the point
where x falls in the uv-plane. If 0