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We don't know that A/(x - 1) = B/(x - 1) -- that's what we're trying to make true! We start by making A equal to B, then using that to show that A/(x - 1) = B/
by panic 8y ago
We don't know that A/(x - 1) = B/(x - 1) -- that's what we're trying to make true! We start by making A equal to B, then using that to show that A/(x - 1) = B/(x - 1). The process goes backwards from how it's written on the page.
For example, say we have functions f_a(t, x) which are equal to x for t > a and 0 for t <= a. To make f_1(x, 10) / (x - 1) = f_1(x, y) / (x - 1), we can solve for f_1(x, 10) = f(x, y) — y = 10 — then divide on both sides, despite the function being discontinuous at x = 1.
- JadeNB 8y agoI think that we are in violent agreement. The logical process could go backwards from how it's written on the page (although it doesn't have to, if sufficient theoretical machinery is in place); but the pedagogical process goes in the direction that it's written on the page. Technically speaking, any class that uses partial-fractions decompositions should prove that they always exist (in which case the procedure that starts by assuming that they exist is justified, and no backwards run is needed), or should finish every such problem by checking that the putative decomposition is valid (the backwards process to which you refer).