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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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Peter Percival

Posts: 1,367
Registered: 10/25/10
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Sep 28, 2013 3:54 PM
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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
>



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9/28/13
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Hetware
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Peter Percival
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