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Topic: Is (t^2-9)/(t-3) defined at t=3?
Replies: 166   Last Post: Oct 30, 2013 9:41 AM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Sep 28, 2013 6:36 PM

In article <9cSdnYGhH7CMpNrPnZ2dnUVZ_sCdnZ2d@megapath.net>,
Hetware <hattons@speakyeasy.net> wrote:

> 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 F(t) is the result of physically measuring two quantities and then
dividing the results, don't count on crossing any bridges built using
that sort of result.
--

Date Subject Author
9/28/13 Hetware
9/28/13 Michael F. Stemper
9/28/13 scattered
9/28/13 Hetware
9/28/13 quasi
9/28/13 Hetware
9/28/13 quasi
9/28/13 Peter Percival
9/29/13 quasi
9/28/13 Hetware
9/28/13 Richard Tobin
9/28/13 Hetware
9/28/13 tommyrjensen@gmail.com
9/29/13 Hetware
10/6/13 Hetware
10/6/13 Peter Percival
10/6/13 Hetware
10/6/13 quasi
10/8/13 quasi
10/7/13 Peter Percival
9/29/13 Michael F. Stemper
9/29/13 Hetware
9/29/13 quasi
9/29/13 Hetware
9/29/13 magidin@math.berkeley.edu
10/6/13 Hetware
10/6/13 magidin@math.berkeley.edu
10/7/13 Hetware
10/7/13 LudovicoVan
10/7/13 Peter Percival
10/8/13 magidin@math.berkeley.edu
10/12/13 Hetware
10/12/13 fom
10/13/13 magidin@math.berkeley.edu
10/13/13 Richard Tobin
10/13/13 Hetware
10/13/13 Peter Percival
10/13/13 fom
10/13/13 magidin@math.berkeley.edu
10/13/13 magidin@math.berkeley.edu
10/8/13 quasi
10/8/13 magidin@math.berkeley.edu
10/8/13 quasi
10/8/13 quasi
10/12/13 Hetware
10/13/13 quasi
10/13/13 Peter Percival
10/9/13 magidin@math.berkeley.edu
10/9/13 fom
10/10/13 magidin@math.berkeley.edu
10/10/13 fom
10/7/13 Peter Percival
10/7/13 Hetware
10/7/13 fom
10/7/13 Peter Percival
9/29/13 quasi
9/30/13 Peter Percival
9/30/13 Peter Percival
9/30/13 Peter Percival
9/30/13 RGVickson@shaw.ca
9/30/13 Roland Franzius
9/30/13 Richard Tobin
9/30/13 RGVickson@shaw.ca
9/28/13 Peter Percival
9/28/13 Hetware
9/29/13 Peter Percival
9/28/13 Virgil
9/29/13 quasi
9/29/13 Virgil
9/29/13 Hetware
9/29/13 quasi
9/29/13 Hetware
9/29/13 LudovicoVan
9/29/13 quasi
9/29/13 Virgil
9/29/13 magidin@math.berkeley.edu
9/29/13 Peter Percival
9/29/13 FredJeffries@gmail.com
9/30/13 Hetware
9/30/13 magidin@math.berkeley.edu
10/6/13 Hetware
10/6/13 Peter Percival
10/6/13 Peter Percival
10/6/13 magidin@math.berkeley.edu
10/6/13 Peter Percival
10/6/13 magidin@math.berkeley.edu
10/6/13 David Bernier
9/29/13 Peter Percival
9/28/13 Hetware
9/29/13 Richard Tobin
9/30/13 Ciekaw
9/30/13 Robin Chapman
9/30/13 Virgil
9/30/13 LudovicoVan
9/30/13 LudovicoVan
10/6/13 Hetware
10/7/13 Robin Chapman
10/7/13 David Bernier
10/7/13 Hetware
10/7/13 LudovicoVan
10/8/13 Hetware
10/9/13 Peter Percival
10/9/13 Richard Tobin
10/7/13 Peter Percival
10/8/13 Hetware
10/8/13 Virgil
10/8/13 Hetware
10/9/13 magidin@math.berkeley.edu
10/9/13 Peter Percival
10/10/13 Ciekaw
10/9/13 Peter Percival
10/10/13 Tim Golden BandTech.com
10/13/13 Hetware
10/13/13 Peter Percival
10/13/13 Hetware
10/14/13 Peter Percival
10/13/13 Hetware
10/13/13 fom
10/13/13 Hetware
10/13/13 fom
10/14/13 fom
10/14/13 Hetware
10/14/13 magidin@math.berkeley.edu
10/14/13 magidin@math.berkeley.edu
10/14/13 Peter Percival
10/14/13 Hetware
10/14/13 quasi
10/16/13 @less@ndro
10/16/13 quasi
10/19/13 Hetware
10/19/13 quasi
10/19/13 Hetware
10/20/13 fom
10/20/13 quasi
10/20/13 Hetware
10/20/13 fom
10/20/13 Hetware
10/20/13 Peter Percival
10/20/13 Richard Tobin
10/20/13 Hetware
10/30/13 @less@ndro
10/19/13 Hetware
10/10/13 Ronald Benedik
10/10/13 Peter Percival
10/10/13 Virgil
10/18/13 Hetware
10/19/13 Peter Percival
10/19/13 fom
10/19/13 Peter Percival
10/19/13 Hetware
10/19/13 Peter Percival
10/19/13 Hetware
10/19/13 fom
10/19/13 magidin@math.berkeley.edu
10/19/13 Hetware
10/19/13 magidin@math.berkeley.edu
10/20/13 Hetware
10/20/13 quasi
10/20/13 quasi
10/20/13 Hetware
10/20/13 Peter Percival
10/20/13 magidin@math.berkeley.edu
10/20/13 Hetware
10/20/13 Arturo Magidin
10/20/13 Hetware
10/20/13 magidin@math.berkeley.edu
10/19/13 fom