MadSci Network: other |
You ask a very good question. The mathematics of infinite can be tricky. For example 0/0 can sometimes approach infinity or some other bounded number.
The key is in the approach to the singularity ( division by zero). The difference is in what you are doing, but typically mathematics does not distinguish between levels of infinity.
L'Hôpital's Rule is a good place to start in the business of infinity. l'Hôpital's rule says that you can take the derivative of the numerator divided by derivative of the denominator to get the value at the division by zero point.
The sinc function (sin(x))/x appears to be zero divided by zero when x = 0, but in reality it evaluates to 1 because of the way it approaches its apparent singularity.
f(x) sin(x) ----- = -------- g(x) x as x ---> 0 f'(x) cos(x) ----- = -------- g'(x) 1 as x ---> 0 cos(x) ---> 1 since cos(0) = 1 cos(x ---> 0) ------ = 1 1 so sin(x) ------ = 1 at x = 0 xanother way to think about it is
sin(x) ~= x - x^3/(3!) + x^5/(5!) -... for x << 1 (x much much less than 1) only the first term is important sin(x) ~= x for x <<1so
sin(x) x - x^3/(3!) + x^5/(5!) -... x ------ = ------------------------------ ~= ----- = 1 x x xRemember: it is all in the approach to infinity!
Sincerely,
Tom "Nearly Finite" Cull
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