Science Forums 2

[Qfwfq]

Originally Posted by Don Blazys

This handy little technique is known as "L'Hopitals rule", and is quicker and more accurate than trying to "approach" x=1 as a "limit".

This is a statement that we mathematicians consider totally inaccurate. Don, de L'Hopital's theorem is about limits. How in the world can it be an alternative to "trying" to approach as a limit?

In any case, it is one way of calculating the limit but there is also an algebric method since numerator and denominator are both plain polynomials; the fact that they have a common zero at implies them having a common factor equal to , therefore division (see Ruffini's method) leaves no remainder:



As you can see, for all and for which it is defined, the fraction I gave has the same value as the one in here:



The fact is that the rhs does have a value at , while the lhs is discontinuous there and so only has a limit but not a value. So, strictly, the above identity is a correct one only under the restriction specified.

Originally Posted by Don Blazys

As for the equation:
a=e^(lna),
it is indeed an identity for all "natural numbers":
a={1,2,3...},
whereas the trivial:
a=a
is an identity for all "non-negative integers":
a={0,1,2...}.

Don, we mathematicians don't get why you are restricting to integers and naturals, neither why you restrict to non-negative even in the case of . Surely you aren't proposing to teach students in this manner?

The identity is correct .

The identity is correct .

Let's forget about the subtleties of complex values! You failed to address the last thing I had asked: does the identity have any bearing on ? As you can see, the domain of validity of the first does not restrict the second one. Likewise can be said about the two polynomial fractions, the discontinuity of the one, wherein it has no value, does not by any means imply the other not being valid for .