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Aug 8, 2012 · The polynomial function in the denominator, even with the very small exponent, will dominate the logarithm function. The denominator will eventually get larger than the numerator and drive the quotient towards zero.
Aug 3, 2023 · The numerator is the top number in a fraction. It represents the number of equal parts we consider from a certain quantity. In contrast, the denominator is the bottom number in a fraction. It represents the number of equal parts a whole quantity is divided into.
The reason is simple: the denominator is growing faster than the numerator, and so dominates, or “wins,” the overall behavior of the fraction, forcing the function to zero as x grows large in either direction.
According to our order of dominance, the denominator is a linear function and should dominate the numerator function, which is a logarithm. Therefore, as approach infinity, this whole fraction should approach .
Quite blindly I've learnt a basic rule about fractions: The Denominator of the Denominator goes to the numerator. I'm confused about it and I'll give an example as to why. Imagine the following: 1/2/2.
Sep 5, 2016 · • Possibility 2: Denominator dominates. If the largest power in the denominator is greater than the largest power in the numerator, then the limit is 0. This rule holds for both $x \to \infty$ and $x \to \, -\infty.$
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The intuition is similar, and it has to do with the asymptotic dominance of the numerator versus denominator. Theorem 1. (Asymptotic Dominance) Suppose that b > 1, p > 0, and a > 1 are real numbers. Then, as n ! 1, we have. (1) log b(n) < np< an< n! < nn where n runs over the positive integers.
