- #1

qspeechc

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## Homework Statement

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]

## 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.