theorem :: LIMFUNC3:46

for x0 being Real
for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,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 f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f . g <= 0 ) ) holds
f ^ is_divergent_to-infty_in x0