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

>

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