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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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quasi

Posts: 10,188
Registered: 7/15/05
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Sep 28, 2013 1:53 PM
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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.

quasi


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9/28/13
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Hetware
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Michael F. Stemper
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quasi
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Hetware
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quasi
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Peter Percival
9/29/13
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quasi
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Hetware
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Richard Tobin
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Hetware
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Hetware
10/6/13
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quasi
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Hetware
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Hetware
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fom
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quasi
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quasi
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10/16/13
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quasi
10/19/13
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quasi
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Hetware
10/20/13
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quasi
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Virgil
10/18/13
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Hetware
10/19/13
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10/19/13
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fom
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Peter Percival
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Hetware
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10/19/13
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