Re: Stephen J Arnett - limit sin[xy]/y
- From: David W. Cantrell <DWCantrell@xxxxxxxxxxx>
- Date: 07 Jan 2008 13:47:34 GMT
"N. Silver" <mathelp@xxxxxxxxxxxxxxxx> wrote:
You have a point (no pun intended). Some texts (Anton,
Strang) support my original point of view; that is, the
function has to be defined in some open interval containing
the limit point in order to qualify, while other texts (Thomas)
require only points in the domain need to be considered,
provided, of course, that they are arbitrarily close to the
limit point in some deleted neighborhood.
I had already pointed out, in responding to Robert Israel in the
parent thread, that texts differ in this matter. Texts are not
infallible, of course. I think it is even possible that Anton, Strang
and company did not actually intend to use such a restrictive
definition.
Until persuaded otherwise, I am of the opinion that the less
restrictive definition of limit, as given in the EoM (to which there
is a link below), is preferrable to the more restrictive type.
I think it is very unfortunate that mathematics lacks a standardizing body.
For a concept as fundamental as that of limit, it is absurd that we should
have any possible disagreement, that there should be any doubt as to what
is standard.
Of course, I am not arguing that one should never be allowed to use a
nonstandard version of a standard concept. Mathematicians should be allowed
to use whatever definitions they wish to use. But if one mentions "limit"
in, say, the context of elementary calculus, there should be absolutely no
doubt what the (single!) standard is. Then, if any people wished to deviate
from that standard, that would be fine. But they would know that their
definition(s), being nonstandard, would have to be stated for the reader.
David
David W. Cantrell wrote:.
N. Silver wrote:
David W. Cantrell wrote:
N. Silver wrote:
Stephen Arnett wrote:
limit sin[xy]/y
(x,y]-->(0,0)
The limit does not exist, because it does not
exist along the path y = 0.
The function itself does not exist for y = 0.
Since the x-axis is not in the domain of the
function, what happens there is irrelevant in this
problem. The desired limit is 0. See my exchange
with Robert Israel in the recent thread "calculating
limit of a trigonometric function".
We probably do not disagree to any great extent.
I'm not sure about that.
Here's my take on it:
Let's return to functions of one variable.
There are only two directions to approach
a limit point. Consider lim(x->0)sqrt(x).
You'll agree, I hope, that the limit does not
exist,
I don't agree with that.
whereas lim(x->0+)sqrt(x) = 1.
I agree that the limit from the right is 0,
which is what you intended to say.
(Note that x < 0 is not in the domain.)
Exactly. Thus, what happens for x < 0 is irrelevant in determining
lim(x->0)sqrt(x). That limit is 0.
As I said in replying to Robert Israel in the previous thread:
FWIW, the definition I use is equivalent to that given in the
third section, "The limit of a function", of
<http://eom.springer.de/L/l058820.htm>.
AFAIK, the analogous situation for real-valued
functions of two variables is that the limit
must exist for every possible path in the plane
to the limit point in some deleted neighborhood.
With the correct understanding of "possible", I agree.
But the given function is undefined along y = 0
for every open ball containing the origin.
And thus any path having y = 0 is _not possible_.
If we examine a text that has a definition, we
may get that the domain of the function must
include ALL points arbitrarily close to the limit
point.
The definition to which I gave a link above does not require that.
David
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