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.

~ 61. 62. Determination of a quotient of the form -. 109
For we have:
p a + bex (x) -- _
2) If we put (p(x) = x + sin x, '(x) = x, we have for x = o,
(il. (p (x} Y,, xre -+ sin x i j. t * f(? '(x) 1 f os + 1
Lirm - Lim { + = 1, but Lim - 1
is indeterminate. Although the function p (x) increases continuously, and
becomes determinately infinite, it does not satisfy either the condition that
qp (x + h) - p (x)
or that cp'(x) have a determinate value for x==oo.
If (x) and 4 (x) both vanish for x = oo (see ~ 60), let us write:
~-r (x) ) ) 1(x),) I = - 1((x); both functions tpi and ij, become oo for
= oo; so by the rule last found
(ix~),(x) Li '(x+ qh) h '(x+el) ((x'+0'h)
X= —a. X.=o-o X=-o X=oo
Hence we have the equation:
f w(x) cp'(x + O'Jh
tjP(x) _ p'(x+eh) l'
X=00 XS=O
But the problem of determining the value is not directly solved by
this equation, since rp' and 4p' must also vanish for x = o (~ 24e); it
may however lead to a simplification.
Example:
a 1
1(1~ a) __ J LU
Z(l+a)} - I
\ x J _x' x1
I 1(~) 2 x ---"+