An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 28. 29.

The exponential function.

53

(I A+ ) > ( + Y) > (+ -t-.) ),
or
(, + ). (1 + ) > (1 + ) > (1 +~ + (, + t)Here both (I + 1 and (I + +- converge to the value e, while
as n increases arbitrarily (1 - — ) and (i + 1 )K have unity for
their limit; thus superior and inferior limit approximate to the value c,
therefore for the present n we also have
Lim (I + e.
Lastly if m is negative, let us put m - u; then
+(I1 --- Ij (1 -__) =_ = (i, 1
- (1 +1 ~ (' +- )
Therefore
Liin (1 + )- ) = Limi (1 + -L_ 1 ~ Lim (1 + - ) e 1.
Accordingly in whatever way Ax may converge to zero we have
-=y a * Lim - a Lim -
log (1 + -) }
- -- ax -e- G ax. elog a.
alog e a
[Hence we see that the exponential function whose base is e, has the
property of reproducing itself unaltered by differentiation; we have
(ex =) ex, since elog e = 1.
The irrational number e is called the base of the natural system of
logarithms; the logarithm relative to this base is briefly denoted by 1.
29. The trigonometric functions.
a) y = sin x.,) y = cos x.
Although we have as yet defined these functions only geometrically
for all finite values of x, the propositions in ~ 12 enable 'us to assign
the derived function of each:
for ) y _ sin (x + Ax) - sin _ 2 sin A Ax cos (xc + 4Ax)
-or AaX x Ax
dy _ Lim sin ( LA ) Lim cos (x + Ax) == cos x.
dx ^/~x A X