theorem :: LIMFUNC3:39

for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f2,x0) <> 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f1 / f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 / f2) ) ) holds
( f1 / f2 is_convergent_in x0 & lim ((f1 / f2),x0) = (lim (f1,x0)) / (lim (f2,x0)) )