A Useful Lemma 
Given f(x) and f'(x) continuous (8.6) 
| f (x)| <M (8.7) 
If'(x)]<M (8.8) 
A 
then im | f(x) dx = a f (0) (8.9) 
~- od WN 
Proof 
nS . € 
lim if SINNX F(x) dx = lim | S8N¥ f(x)dx+ lim | SONX £(x) dx 
N—co a : N—->co aN N> Le 
stim fa f (x)dx (8.10) 
€ 
A 
[eG )ax 
€ 
A 
A 
lim {| ae f(x) dx | < lim | - 49 <9 | [+ tim 
€ aon € 
N-~ oo N->co 
< lim (24 +204) <0 (8. 11) 
m™ 
EN 
lim] S85% £(x)dx= lim] 5% Bin. x. filer “)dx' = Jim 9 [88x m f(o) (8.12) 
a 
Nooo) Neo J 
Where x=Nx (8.13) — 
Plate XXIV 
SoS 
