Question: Find lim h->0 sin(π(x+h)- sin(πx)/ h .
Use l'Hôpital's rule.
L = lim [h->0] u/v = lim [x->0] u'/v'
where
u = sin(π(x+h)) - sin(πx), u' = du/dh = π.cos(π(x+h))
v = h, v' = dv/dh = 1
Lim [h->0] u'/v' = π.cos(π(x+(h->0)))/1 = π.cos(πx)
Answer: Limit = π.cos(πx)