For example:

$f(n) = n(log_2 n)^4$ vs $g(n) = n^{1/3}$

Does f(n) grow slower than g(n) as a general rule, even for decimal exponents? I tried doing a limit test but symbolab limit calculator is saying that Steps are currently not supported for this problem.

I graphed them over a large range and it does appear that g(n) grows asymptotically faster.


I think you may be confused about the definition of Big O - graphing f(n) and g(n) shows f(n) surpassing g(n) at nearly every scale.

O(f(n)) > O(g(n) because g(n) is a fractional power polynomial, which is by definition sublinear (if you linearly plug numbers into a square root, cube root, etc., the rate at which the output increases decreases with each successive computation). g(n) is a "n log star n" polynomial meaning it follows the form of a log to a power multiplied by n. The rate of growth of these types of functions far exceeds fractional polynomials and its not hard to see why - n is a term of the function, so we know its at least linear.

Here's a helpful resource: https://en.wikipedia.org/wiki/Time_complexity#Table_of_common_time_complexities

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