On 9/28/2013 3:59 PM, quasi wrote: > Hetware wrote: >> quasi wrote: >>> Hetware wrote: >>>> >>>> I'm reading a 1953 edition of Thomas's Calculus and >>>> Analytic Geometry. >>>> >>>> In it he states that given: >>>> >>>> F(t) = (t^2-9)/(t-3) >>>> >>>> F(t) = (t-3)(t+3)/(t-3) = t+3 when t!=3. >>>> >>>> But F(t) is not defined at t=3 because it evaluates to 0/0. >>>> >>>> If someone were to ask me if (t^2-9)/(t-3) is defined when t=3, >>>> I would say it is >>> >>> Then you would be wrong. >>> >>>> because it can be simplified to t+3. >>> >>> To get that result, you had to cancel the common factor t-3 in >>> numerator and denominator. But that cancellation depends on the >>> simplification >>> >>> (t - 3)/(t - 3) = 1 >>> >>> which is only valid if t != 3. >>> >>> In a first level algebra course (Elementary Algebra) where >>> function concepts are not yet in play, the simplification >>> >>> (t^2 - 9)/(t - 3) >>> >>> = (t + 3)(t - 3))/(t - 3) >>> >>> = t + 3 >>> >>> is allowed, without worrying about exceptional values of t for >>> which the simplification fails. >>> >>> But at the next level of algebra, algebraic expressions are >>> often being regarded as functions, so more care is taken to >>> identify those exceptional values. >>> >>>> Am I (and/or Thomas) engaging in meaningless hair-splitting >>>> regarding the question of F(3) being defined? >>> >>> For functions, identifying the precise domain is key. >>> >>> Thomas is correct. >>> >>> Those hairs _need_ to be split. >>> >>> All modern precalculus and calculus texts are careful (in the >>> context of deciding whether functions are equal) to identify >>> exceptional values where simplifications fail. >> >> The reason I have some misgiving about saying that F(3) is >> undefined is > > F is being regarded as a _function_ not just an expression, > so the definition of F is taken literally. You don't have the > right to simplify the definition before evaluating the > function unless you can guarantee that the results would > always be the same, before and after. > > Thus if F(t) = (t^2 - 9)/(t - 3), then by direct substitution > > F(3) = (3^2 - 9)/(3 - 3) = 0/0 > > which is undefined.
I understand why it is considered undefined.
> If we let G(t) = t + 3, then G(3) = 3 + 3 = 6. > > Thus, F and G are not equal as functions. > > They _are_ equal for all values t _except_ t = 3. > > Thus the graphs are not the same. > > The graph of the equation y = G(x) is a straight line. > > The graph of the equation y = F(x) is the same straight > line, but with a missing point at (3,6), usually symbolized > by placing a small open circle around (3,6) to make clear > that there is a missing point there (a "hole"). > > With regard to continuity, the function G is continuous, > whereas the function F has a discontinuity at t = 3. > > However since the limit of F, as t approaches 3 _exists_, > we say that F has a _removable_ discontinuity at t = 3. > > You need to drop your preconceptions on this so as to make > progress in your study of Calculus. All modern Calculus texts > are consistent with regard to this issue. > > quasi >
So the answer is consensus among mathematicians holds that F(t) = (t^2 - 9)/(t - 3) is undefined at t=3? Perhaps what I should have said at the outset is something along the lines of: on any given day, if I'm setting up an equation in physics, and produce an expression such as F(t) = (t^2 - 9)/(t - 3), I treat it as t+3, and do not expect any adverse consequence from doing so.
If I conceive of mathematics as an exercise in defining and manipulating symbols, it seems that declaring constructs such as F(t) = (t^2 - 9)/(t - 3) to be undefined at t=3 is arbitrary. The fact that there is an obvious candidate for a value of F(t) at t=3 tells me that accepting that candidate as the value at t=3 does not contradict the definition of a single valued function of one variable.