Limits of Trigonometric Functions

Direct substitution for sine and cosine. The functions $\sin x$ and

$\cos x$ are defined for every real number $x$ (the unit-circle definition covered earlier)

and their graphs are unbroken, continuous curves with no jumps or gaps anywhere. Because of

this, for any real number $a$,

-- direct substitution again, exactly as for polynomials. Since $\tan x=\sin x/\cos x$, the

quotient rule of Section 2 then gives $\displaystyle\lim_{x\to a}\tan x=\dfrac{\sin a}{\cos a} =\tan a$, valid whenever $\cos a\neq0$, i.e. whenever $a$ is not an odd multiple of $\pi/2$.

A standard limit (stated, not re-derived here). The single most important trigonometric

limit in this chapter -- used to differentiate $\sin x$ from first principles in Section 10, and

throughout physics and engineering -- is

This result is stated and used here as a standard, previously-established limit (its full

proof compares the areas of a triangle, a circular sector and a larger triangle enclosing an

angle $x$, and belongs to a geometric treatment of the circle rather than to algebra of limits).

A useful companion follows immediately from it:

using $\cos0=1$ and the product rule of Section 2.

Reducing $\sin(kx)/x$ to the standard form. For any nonzero constant $k$,

by writing $x=\dfrac{kx}{k}$ and noting $kx\to0$ as $x\to0$. This same trick -- scaling the

argument to match the denominator exactly -- is the standard method for any limit of the form

$\sin(kx)/x$, $\tan(kx)/x$, or a ratio $\sin(px)/\sin(qx)$ (rewrite each as a multiple of its

own standard limit and take the ratio of the constants).

The limit $(1-\cos x)/x^2$. Using the half-angle identity $1-\cos x = 2\sin^2(x/2)$, …