Compare function growth rate, dominance, and Big O behavior as x approaches infinity
Growth rates determine how functions behave as input becomes very large. This function growth rate calculator compares polynomial, logarithmic, and exponential functions using the limit of a ratio and reports which function dominates. This analysis is common in algorithm efficiency, Big O notation, asymptotic behavior, and long-term trend comparison.
To compare growth rates, compute the limit of f(x)/g(x) as x->inf. If the ratio -> 0, g dominates f. If -> inf, f dominates g. If -> constant, they grow at the same rate. The hierarchy is absolute: exponential functions always outgrow polynomial functions eventually.
For two functions f(x) and g(x), divide one by the other and take the limit as x goes to infinity. A zero limit means the denominator grows faster, an infinite limit means the numerator grows faster, and a nonzero constant means the functions have the same asymptotic growth rate.
Higher degree dominates: x^4 grows faster than x^2. Ratio x^4/x^2 = x^2 -> inf. Same degree: ratio = constant.
Exponential always wins: 2^x / x^100 -> inf. Regardless of polynomial degree, exponential eventually dominates.
Any positive power of x dominates: x^0.0001 / ln(x) -> inf. Polynomial grows faster than any log.
f=O(g): f <= c*g for large x. f=Theta(g): same growth. f=o(g): f/g -> 0 (f dominated).
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