Why does the book say if f(x) is continuous at a then lim ( f(a + Δx) - f(a) ) / Δx, that Δx will go to zero. What does that mean geometrically? Δx→0 More importantly, why would Δx not approach zero if f(x) is not continuous at a? Im guessing it has something to do with the slopes of the secant lines approaching a definite number therefore Δx will approach 0. However say we were given g(x) = 1/(x - a) Then: g'(a) = lim ( 1/(Δx) - 1/0 ) / Δx therefore not only is 1/0 not defined but 1/Δx does not have a limit near 0 in Δx→0 this case so the limit as Δx → 0 does not exist. Am I on to something or am I way off base?