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Feb 20, 2016 · The answer is yes, although in some cases (like the one you have given) it takes a very long time for the polynomial function to catch up to and ultimately dominate the log function. A rigorous formation of what you are saying is: $$ \lim_{x \to \infty} \frac{\log(x)}{P(x)}=0$$
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Aug 8, 2012 · Look farther out and farther up, the exponential dominates and will eventually lie above the polynomial (after x = 7.334). Here’s an example that pretty much has to be done using the dominance approach. The polynomial function in the denominator, even with the very small exponent, will dominate the logarithm function.
Jul 8, 2024 · $\begingroup$ That's a squishy statement, but they likely mean any power of logarithms, compositions of logarithms, and combinations thereof. Note that you can't say that this is true for any expression involving logarithms as it fails for something like $\log(x)^{\log(x)}$. These things are mostly heuristics and not meant rigorously. $\endgroup$
We can summarize our discussion of dominance from the preceding several screens as: \[ a^x \text{ dominates } x^r \text{ dominates } \ln x \] In words: exponentials dominate power functions (and polynomials) dominate logarithms.
2. Log versus Polynomial To prove the polynomial dominance over the logarithm, we are looking at the limit lim x!1 log b (x) xp where b>1 (otherwise the logarithm would be a decreasign function) and p>0 (otherwise the polynomial would be decreasing of the type 1=xp). To compute the above limit, notice
Exponential functions are functions of the form \( f(x) = a^{x} \), where the base a is a positive real number. These functions grow much faster than polynomial and logarithmic functions. For example, the function \( e^{x} \) (where e is Euler's number) grows exponentially and dominates any polynomial function.
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Subsection 0.6.2 Polynomial Functions Power functions have very predictable behavior but when we add or subtract several power functions we can model much more complicated behavior. A function made out of the sum of several power functions is known as a polynomial. Polynomial Functions. A polynomial function is a function of the form
