```Date: Sep 28, 2013 3:54 PM
Author: Peter Percival
Subject: Re: Is (t^2-9)/(t-3) defined at t=3?

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_With what domain and range?> 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.>> 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>-- The world will little note, nor long remember what we say hereLincoln at Gettysburg
```