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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: 10,226
Registered: 7/15/05
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 19, 2013 3:57 PM
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Hetware wrote:
>
>Let me propose a test question:
>
>Given
>
> f(x) defined over R,


> limit[f(s), s->x] exist over R,

> and g(x) = limit[f(s), s->x]
>
>can g(x) be used to find the value of f(x) for all x in R?


No, not unless it is also given that f is continuous.

For example, define f by

f(x) = 1 if x != 0

f(0) = 2

Then

f(x) is defined for all x in R,

and limit (s -> x) f(s) exists for all x in R,

but if g is defined by

g(x) = limit (s -> x) f(s)

then g(x) = 1 for all x in R, hence

g(0) != f(0)

so f,g are not the same function.

Now suppose f(0) is known to exist, but the value of f(0) is
not explicitly specified. Then f(0) could potentially be any
real number. In particular, in the absence of more information,
you can't conclude that f(0) = g(0).

On the other hand, suppose your proposed question is reworded
in the following way ...

Given:

f(x) is defined and continuous for all x in R.

Then if g is defined by

g(x) = limit (s -> x) f(s)

can g(x) be used to determine the value of f(x) for all x in R?

The answer is yes -- in fact, f(x) = g(x) for all x in R.

quasi


Date Subject Author
9/28/13
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Hetware
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Michael F. Stemper
9/28/13
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Hetware
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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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Hetware
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Hetware
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Michael F. Stemper
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Hetware
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quasi
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Hetware
9/29/13
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magidin@math.berkeley.edu
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Hetware
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magidin@math.berkeley.edu
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Hetware
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Hetware
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fom
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Richard Tobin
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Hetware
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fom
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quasi
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quasi
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quasi
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fom
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fom
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quasi
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Peter Percival
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quasi
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Virgil
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quasi
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10/16/13
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quasi
10/19/13
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Hetware
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quasi
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Hetware
10/20/13
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fom
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quasi
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Hetware
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Hetware
10/30/13
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10/19/13
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Hetware
10/10/13
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Virgil
10/18/13
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Hetware
10/19/13
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Peter Percival
10/19/13
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fom
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Peter Percival
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Hetware
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Hetware
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fom
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magidin@math.berkeley.edu
10/19/13
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Hetware
10/19/13
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10/20/13
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quasi
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