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#1 2007-03-22 22:33:20

mikau
Member
Registered: 2005-08-22
Posts: 1,504

non elementary functions

can some non elementary functions be expressed as the limit of a ratio of two elementary functions?

In otherwords, if P(x) is a non elementary function, and g(x) and f(x) are both elementary functions, is it possible that:

P(x) = limit f(x)/g(x) as x approaches "a"

at least in some scenarios?


A logarithm is just a misspelled algorithm.

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#2 2007-03-23 02:12:20

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: non elementary functions

P(x) = limit f(x)/g(x) as x approaches "a"

Do you mean:

I don't see any reason why not.  Might be hard to come up with an example though.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2007-03-23 05:59:39

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: non elementary functions

Not necessarily P(a), rather, P(x) = lim f(x, h) / g(x, h) as h approaches a.

say f(x, h) = x^2 +2hx + h^2 and g(x, h) = ln(x) +he^x , if you take the ratio of f(x,h)/g(x,h) as h approaches zero from the right, the ratio approaches x^2/ln(x). In this case, we can find the limit and it is elementary, but in some cases, we can't find what the exact limit is.

Obviously, if this limit approaches P(x) which is nonelementary, the limit would have to be one of those limits we can't solve, or the function would be elementary.

I wish i knew more about non elementary functions. I mean, couldn't e^x be considered non elementary until we decided to program its series approximations into our calculators?


A logarithm is just a misspelled algorithm.

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#4 2007-03-23 06:12:23

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: non elementary functions

Not necessarily P(a), rather, P(x) = lim f(x, h) / g(x, h) as h approaches a.

Ok, how you have a function, lim f(x, h) / g(x, h) as h approaches a, of three variables, and P(x) is a function of 1.  Obviously, this function can't be well defined (i.e. it's not a function) because you are allowed to vary three things as opposed to 1.

say f(x, h) = x^2 +2hx + h^2 and g(x, h) = ln(x) +he^x , if you take the ratio of f(x,h)/g(x,h) as h approaches zero from the right, the ratio approaches x^2/ln(x).

Ok, here I take it that a = 0.  What if a = 1?  Then P(x) (when a = 0) is not the same as P(x) (when a = 1).


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2007-03-23 10:53:19

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: non elementary functions

I was using "a" as some unspecified constant. Not a variable.

for instance, P(x) = f(x, h)/g(x, h) when and only when h approaches zero. In that case, then P(x) can be found by inserting x into the functions g and f and letting h approach zero. If you let h approach some other number then the relationship doesn't hold.

but it doesn't necessary have to be a ratio of two functions.

The point is this. We could let f(x, h) = (1/h + 1)^hx.

We could then define e^x as the limit of  f(x, h) as h approaches infinity.  We could have used an infinite series to find e^x but here we managed to find it as a limit.

Likewise, a lot of non elementary functions can be expressed as an infinite series, so could some non elementary functions be expressed as a simple limit instead of an infinite series or riemann sum?


A logarithm is just a misspelled algorithm.

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