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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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Hetware

Posts: 148
Registered: 4/13/13
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Sep 29, 2013 6:47 PM
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On 9/29/2013 8:45 AM, Michael F. Stemper wrote:
> On 09/28/2013 04:20 PM, Hetware wrote:
>> On 9/28/2013 4:24 PM, Richard Tobin wrote:
>>> In article <9cSdnYGhH7CMpNrPnZ2dnUVZ_sCdnZ2d@megapath.net>,
>>> Hetware <hattons@speakyeasy.net> wrote:

>
>>>> So the answer is consensus among mathematicians holds that F(t) =
>>>> (t^2 -
>>>> 9)/(t - 3) is undefined at t=3?

>>>
>>> Yes.
>>>

>>>> 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.

>>>
>>> Your simplification is not valid for t=3.

>
>> I believe most mathematicians solving for x as a function of t given
>>
>> t^2 - 9 = x (t - 3)
>>
>> would not hesitate to factor the left hand side and divide both sides by
>> t - 3 without treating t = 3 as a special case.

>
> This is high-school stuff. Literally.
>
> I just pulled my text[1] from Algebra II (tenth grade) off the shelf.
>
> Turning to Section 3.6, "Rational Expressions: Reduction to Simplest
> Form", I see:
>
> It should be observed that since division by zero is excluded
> as an operation, a fraction whose denominator is zero has no
> meaning. For example, the fraction 9/(y-7) has no meaning
> when y=7. Also, the fraction (x+5)/(x^2-9) has no meaning
> when x=3 or x=-3.
>
> We sometimes encounter fractions such as (x-3)/(3-x)
> (x!=3). [...]
>
> The authors then proceed to work through several examples of fractions
> with polynomials in the numerator and denominator. In every case, when
> the fraction is specified, they include:
>
> where (d(x)!=0)
>
> (I'm using d(x) here to represent whatever the denominator polynomial
> happens to be.)
>
> If your high school math classes neglected to point out that this
> exclusion is necessary, they were faulty. However, the errors and
> omissions of your past teachers do not change the fact that it is.
>
>
> [1] _Algebra and Trigonometry: A Modern Approach_, Peters & Schaaf,
> van Nostrand, (c) 1965


What I am saying is that if I encountered an expression such as
(t^2-9)/(t-3) in the course of solving a problem in applied math, I
would not hesitate to treat it as t+3 and not haggle over the case where
t = 3.

If I wanted to express x as a function of t given (t^2-9) = x(t-3), I
would have no compunction about "dividing out" t-3. Since I can arrive
at the same result through a different method, I do not see why treating
(t-3)/(t-3) = 1 in this context is an error.

When I do something like

(t^2-9) = x(t-3) <=> (t^2-9)/(t-3) = x(t-3)/(t-3) <=> (t^2-9)/(t-3) = x

I don't feel a twinge of unease. I have no experience of it leading to
an incorrect final result. If I encounter something of the form
f(t)/(3-t) and the problem domain includes t = 3, I anticipate a problem
at t = 3. If I can fiddle with f(t) and factor out 3-t, I simply cancel
them out, and keep going.


Date Subject Author
9/28/13
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Hetware
9/28/13
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Michael F. Stemper
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9/29/13
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Hetware
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Richard Tobin
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Hetware
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9/29/13
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Hetware
10/6/13
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Hetware
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