Science Forums 2

[Don Blazys]

To:Qfwfq.

When I wrote that applying L'Hopitals rule is quicker and more accurate than trying to "approach" x=1 as a "limit", I simply meant that it is easier to evaluate the expression involving derivatives at x=1 than it is to evaluate the original expression at x=1.00000000001 or x=.99999999999 or some such approximation to 1.

I certainly didn't mean to imply that applying L'Hopitals rule automatically allows us to dispose of the notion of limits. (I might be crazy, but I'm not that crazy!)

Now, the question is, at x=1, does 0/0=1.25?

I say yes, because I don't subscribe to the view that at x=1, the indeterminate form 0/0 necessarily constitutes a "discontinuity". For me, the result 0/0 is absolutely consistent with the result 1.25, so the two results can, in that particular case, be set equal to each other.
In other words, from my point of view, if one method of evaluation at x=1 yields 0/0, while another, equally valid method of evaluation at x=1 yields 1.25, then maintaining consistency actually requires that the two results be regarded as "interchangeable" in that particular case.

As for why I restrict the variable "a" in the identity a=a to non-negative integers, well, I am of the opinion that the restriction is both "built in" and "natural".
You see, non-negative integers are the only numbers that can possibly be represented using essentially one symbol such as "a". (All other numbers require some operation as part of their representation).
Now, since every imaginable and reasonable number can be derived by performing operations on non-negative integers, if we reserve the variable
"a" to represent non-negative integers only, then we can derive all other numbers by performing operations on "a".
Thus, if we wish to represent "negative integers", then we should do so by writing "(-a)" rather than just "a".
By the same token, if we want to represent "unit fractions", then we should write "1/a" instead of just "a".
"Negative unit fractions" are better represented as "(-1/a)" than simply by "a", and so on.
It's just a matter of being concise, precise and rigorous, not to mention logical and artistic.
I am well aware of the fact that this amount of rigor is quite "impractical" and "unnecessary" for most everyday problems in mathematics.
Therefore, I would not present the above facts to students who have no interest in my research.

Don.