DERIVATES 
495 
Example 1. f (ж) — x sin 
x Ф 0 in (— 1, 1) 
Here for x = 0, 
Hence 
Example 2. 
Here for 
Hence 
Example 3. 
Here 
= 0 , 
n • 1 
h sm- 
Af = A 
Ax h 
«/'(0) = +1 , 
Lf\ 0)= + l , 
7'(°)= + i , 
= o. 
= sin T . 
h 
Bfc o) = -i, 
Lf\ 0)=-l, 
/'(0)= —I- 
/(ж) = ж* sin - , x Ф 0 in ( — 1, 1) 
X 
x — 0. 
. 1 
л - Sin 7 
A/ = A. 
Ax 
= 0 
ж = 0 
Д/Ч°)= + 20 
Д/'(0)= + оо 
7'(0) = + oo 
/(ж) = ж sin - 
X 
= ж 3 sm 
Л 3 
^/(°)= - ° 0 » 
i/'(0)=- <», 
/'(0)= - GO. 
for 0 < ж < 1 
, for — 1 < ж < 0 
= 0 , 
Д/'(0)= + 1 
Г/'(0)= + оо 
/'(0)= + c© 
ж 
for ж = 0. 
Д/'(0) = -1, 
£/'( 0)=-оо, 
/'(0)=-оо. 
500. 1. Before taking up the general theory it will be well 
for the reader to have a few examples in mind to show him how 
complicated matters may get. In I, 367 seqwe have exhibited 
functions which oscillate infinitely often about the points of a set