by gill1109 » Sun Jun 30, 2019 9:56 am
Joy Christian wrote:Heinera wrote:When one takes a limit, one of course does not have the same variable on both sides of the arrow.
Well, one lives and learns. Here is your opportunity to learn. And the same goes for the other objectors. One can most certainly have the same variable on both sides of the arrow.
We would love to learn. So please teach us! (None of us have ever seen anything like this before).
Let's make things as simple as possible. Let's think of real functions of real variables. What could you mean by writing lim_{x -> g(x)} f(x)? Well, one might interpret the arrow ("converges to") as "gets closer and closer to". Then we could rewrite the limit in conventional notation (with the intended conventional meaning) as lim_{x - g(x) -> 0} f(x). The limit would be the number y, say, if, for every epsilon > 0, there exists a delta > 0, such that if x is such that |x - g(x)| <= delta, then |f(x) - y| <= epsilon.
Well, if that is what you mean, then you can now start the work of figuring out whether or not the two expressions you have given do have limits in this sense, and if so, what they are. Also you can now start investigating whether or not the usual results about products of limits and the like will still be valid in your new notion of limit.
I understand that you furthermore want to couple the dummy variables in both limits to one another. I've never seen anything like that before in 50 years in the best university mathematics departments. Among my teachers were Stephen Hawking, John H . Conway, David Kendall, Peter Whittle; and I got a 1st class degree with distinction. So: please teach us! It is wonderful to see the envelope of what one can do in mathematics being pushed to dizzy new heights.
[quote="Joy Christian"][quote="Heinera"]When one takes a limit, one of course does not have the same variable on both sides of the arrow.[/quote]
Well, one lives and learns. Here is your opportunity to learn. And the same goes for the other objectors. One can most certainly have the same variable on both sides of the arrow. :)
[/quote]
We would love to learn. So please teach us! (None of us have ever seen anything like this before).
Let's make things as simple as possible. Let's think of real functions of real variables. What could you mean by writing lim_{x -> g(x)} f(x)? Well, one might interpret the arrow ("converges to") as "gets closer and closer to". Then we could rewrite the limit in conventional notation (with the intended conventional meaning) as lim_{x - g(x) -> 0} f(x). The limit would be the number y, say, if, for every epsilon > 0, there exists a delta > 0, such that if x is such that |x - g(x)| <= delta, then |f(x) - y| <= epsilon.
Well, if that is what you mean, then you can now start the work of figuring out whether or not the two expressions you have given do have limits in this sense, and if so, what they are. Also you can now start investigating whether or not the usual results about products of limits and the like will still be valid in your new notion of limit.
I understand that you furthermore want to couple the dummy variables in both limits to one another. I've never seen anything like that before in 50 years in the best university mathematics departments. Among my teachers were Stephen Hawking, John H . Conway, David Kendall, Peter Whittle; and I got a 1st class degree with distinction. So: please teach us! It is wonderful to see the envelope of what one can do in mathematics being pushed to dizzy new heights.
