1. The problem statement, all variables and given/known data The problem is from Stewart, Appendix G, A58, no.45. Suppose that F, G, and Q are polynomials, and: F(x)/Q(x) = G(x)/Q(x) for all x except when Q(x) = 0. Prove that F(x) = G(x) for all x. [Hint: Use Continuity] 3. The attempt at a solution I thought the statement was obvious, but ofcourse it isn't if they want you to prove it. This is under the section "Integration of Rational Functions by Partial Fractions", so I guess the proof must somehow use this technique. I guess we know it is true for real numbers a, b, c, that if a/c = b/c then a = b. We just haven't proved it for functions, and I think that is what this is about- am I correct? So the proof needs to be based on the fact that it is true for real numbers- is this correct? That in mind, I decided to take the limit as x approaches a (except if Q(a) = 0) on both sides (sorry about not using Latex- I am in a rush): lim [F(x)/Q(x)] = lim [G(x)/Q(x)] lim[F(x)]/lim[Q(x)] = lim[G(x)]/lim[Q(x)] And because they are all polynomials: F(a)/Q(a) = G(a)/Q(a) Now these are all real numbers, and thus F(a) = G(a). This seems a little dodgy to me, because it does not hold for a when Q(a) = 0, and the questions says prove for all a. Also, we say, when we have: F(a)/Q(a) = G(a)/Q(a) that F(a), Q(a), G(a) are all real numbers, but the answer claims they are functions all-of-sudden! Help is much appreciated.