Special Limits

Definition With $x$ measured in radians,

If $x$ is measured in degrees, the same quotient tends to $\pi/180$ instead, because $x$ degrees equals $\pi x/180$ radians.

Definition For $0 < x < \frac{\pi}{2}$, place the angle $x$ at the center $O$ of the unit circle, with $A = (1, 0)$, $P = (\cos x, \sin x)$ and $T = (1, \tan x)$. The triangle $OAP$ lies inside the circular sector $OAP$, which lies inside the triangle $OAT$. Both triangles have base $OA = 1$, with heights $\sin x$ and $\tan x$. The sector is the fraction $x/(2\pi)$ of the whole disk, whose area is $\pi$, so its area is $x/2$. This is where radian measure enters. Comparing areas gives

Dividing by $(\sin x)/2 > 0$ and taking reciprocals gives

It remains to see that $\cos x \to 1$. For $0 < x < \frac{\pi}{2}$ we have $0 < \cos x < 1$ and $0 < \sin x < x$, so

and squeezing gives $\lim_{x \to 0^+} \cos x = 1$. The squeeze theorem, in its right-hand form, then gives

Both $\cos x$ and $(\sin x)/x$ are even, so the left-hand limits are also $1$, and the two-sided limits follow.

Definition With $x$ in radians,

For $x \neq 0$ close to $0$ we have $1 + \cos x \neq 0$, and

which tends to $1 \cdot 0/2 = 0$ by the product and quotient laws.

Definition For the natural exponential and logarithm,

More generally, for any base $b > 0$,

These limits are not proved here, because a proof needs a precise definition of $b^x$ for irrational $x$. This course follows the common textbook route and takes the first limit as the defining property of $e$. Among all bases $b > 0$, the number $e \approx 2.71828$ is the one for which the limit of $(b^h - 1)/h$ at $0$ equals $1$. The other two limits then follow. For the logarithm, take $x > -1$ with $x \neq 0$ and put $u = \ln(1 + x)$, so that $x = e^u - 1$ and

where $u \to 0$ and $u \neq 0$ as $x \to 0$ with $x \neq 0$. That step uses the continuity of $\ln$ at $1$, which the continuity unit makes precise. For a general base, $b^h = e^{h \ln b}$. Texts that define $\ln x$ as an integral prove all three limits instead, so check which definition a book uses before calling one of them a theorem.

Definition To use a special limit, rewrite the expression so that the exact pattern appears. We know that $(\sin u)/u \to 1$ as $u \to 0$. If $k \neq 0$ is a constant, then $u = kx$ tends to $0$ as $x \to 0$ and $u \neq 0$ when $x \neq 0$, so

The same substitution works for the other special limits. Multiplying and dividing by the same nonzero quantity does not change the expression.

Definition A derivation is circular if it uses a fact whose own proof depends on the result being derived. The standard order is this. First the geometric inequality $\sin x < x < \tan x$ for $0 < x < \frac{\pi}{2}$, then $\cos x \to 1$ and $(\sin x)/x \to 1$ as $x \to 0$ by squeezing, then $(1 - \cos x)/x \to 0$, and only after that the derivatives $\frac{d}{dx} \sin x = \cos x$ and $\frac{d}{dx} \cos x = -\sin x$. The sector area $x/2$ comes from the area of the unit disk and the definition of radian measure. Fully rigorous treatments define sine and cosine by arc length or by power series to avoid relying on pictures.