Thanks left that other site.
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Posted on 02/20/2015 6:01:20 PM PST by LibWhacker
If f and g are differentiable in the open interval containing L [which may be a finite limit or ±∞] and if
limitx→L f'(x)/g'(x) exists,
then the indeterminate form:
limitx→L f(x)/g(x)
where f and g are both zero, or f and g are both ±∞ also exist, and
limitx→L f(x)/g(x) = limitx→L f'(x)/g'(x)
So, just for example: with f(x) = x2 g(x) = 3x2. Both are differentiable, limitx→0 f(x)/g(x) = x2/3x2 which → 0/0, an indeterminate form.
Differentiate twice: limitx→0x2/3x2 = limitx→0 2x/6x = limitx→0 2/6 = 1/3.
Obviously, you could get this answer just by "factoring out" x2. Just algebra; no Calculus required.
However, you can't factor this one: limitx→0 sin(x)/x.
L'Hospital's Rule gives:
limitx→0 sin(x)/x = limitx→0 cos(x)/1 = 1.
Remember to apply L'Hospitals Rule: you don't do the rule for differentiating a quotient.. That would give [f'(x)g(x) - g'(x)f(x)]/[g'(x)]2. You simply take f'(x)/g'(x) and check the limit.
As long as f, f', f'' and g, g', g'' [etc] are still differentiable and their quotient is indeterminate, you can apply the rule as many times as necessary to get an answer.
Can you tell me more about bijective functions, please?
Also, the fact that Zero leads to infinity, can be seen with Obama and our national debt...
Cheers!
What you said...
bttt... looking forward to reading more of this thread. :)
I think I’ll stay well outside this one.
So the upshot is that a 1-1+onto or bijective function has a unique inverse.
Thus this is a way of extending the ordinary notion of counting elements to infinite sets. When we count finite collections, we are putting them into 1-1 correspondence with a subset of the integers. To extend that notion to infinite sets, two sets have the same cardinality or "size" if there is a bijection between them.
The cardinality of the even integers is the same as the cardinality of the integers. Why? Here is a bijection f(N) = 2N.
Every non-empty open subinterval of the real line, no matter how small, has the same cardinality as the whole real line. Why? Here is a bijection: f(x) = arctan(αx); with "α" some suitable scaling factor that maps the arbitrary interval into (-π/2, π/2).
To prove the reals do not have the same cardinality as the integers, produce an enumeration of the reals, then show there is always a real number it doesn't contain. That's Cantor's Diagonalization Theorem.
Here's another way, more abstract but actually less difficult. Define the powerset of a set to be the set of all subsets of a set. So the powerset of {1, 2} is the set {{1}, {2}, {1, 2}, {}} [It's called the powerset because if a finite set has "S" elements, the set of all its subsets has 2S elements.]
Show that there is no bijection between any set and its powerset. Cantor did this already. It's the so-called "who shaves the barber" proof. Then show that there is a bijection between the reals and the powerset of the integers. Since there's a bijection between the reals and the powerset of the integers, there can't be one between the reals and the integers themselves.
In this extended sense [that there is no bijection] there are "more" reals than there are integers.
“Computer models”
Guess that solves it.
Exactly - goes hand-in-hand with Eternity. And the One who actually comprehends it all was wont to say, "I AM".
“”When I was five years old... where does Space end?””
When I was eleven or twelve (late bloomer, I suppose) I asked my Dad, “If space is continuously expanding, what is it expanding into?”
He looked at me with that same “CO form Chicago” expression, rolled his eyes at my Mom and I don’t think he answered my question either.
As a seventh grader, my speculation was that it was expanding into whipped cream. At my present age of 63 I suspect whipped cream isn’t the answer. But just being able to ask the question reminds me how “big” our Creator is and just that thought is more fun than whipped cream.
http://www.youtube.com/watch?v=iCrvibgo1LM
Thanks left that other site.
Could the answer be that the Universe isn’t expanding ‘into’ anything, it is expanding ‘from’ the start.
If there is a smallest unit of space and time and matter and energy then there is no divide by zero problem and therefore no infinity problem.
If the range of the electromagnetic force is not infinite, then it's something. What? If the lifetime of a photon is not infinite, then there are no such things as eigenstates of the Hamiltonian for electromagnetic systems. If there are no stable eigenstates, they have a lifetime. How long?
Claiming that the lifetime of a ground state is some number -- which we must discover -- is not an improvement on saying it's infinite. Its actual value might be significant of some real physics, or it may simply be an environmental value. In the latter case, we would not know that until we have searched for years or decades. Be careful what you wish for.
Max Tegmark should be reminded of this quote.
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