f=O(g): f 0. f=Theta(g): same growth rate.

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Function Growth Rate Analyzer

Compare growth rates of functions as x approaches infinity

Select Functions to Compare

x^n vs x^m e^(ax) vs x^n x^n vs ln(x) e^(ax) vs ln(x)

Comparex^ vsx^

Growth Rate Hierarchy

Slowest: constant < log x < x^0.5 < x < x^2 < x^3

Fast: 2^x < e^x < 10^x < x! < x^x

Exponential dominates ANY polynomial

Polynomial dominates ANY logarithm

Growth rates determine how functions behave as input becomes very large. This analysis is critical in computer science (algorithm efficiency), physics (asymptotic behavior), and economics (long-term trends). The hierarchy is: constant < log < polynomial < exponential < factorial.

⚠ Exponential functions grow faster than ANY polynomial, regardless of degree. Similarly, any positive power of x grows faster than ANY logarithm.

What Is Growth Rate 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.

Polynomial vs Polynomial

Higher degree dominates: x^4 grows faster than x^2. Ratio x^4/x^2 = x^2 -> inf. Same degree: ratio = constant.

Exponential vs Polynomial

Exponential always wins: 2^x / x^100 -> inf. Regardless of polynomial degree, exponential eventually dominates.

Polynomial vs Log

Any positive power of x dominates: x^0.0001 / ln(x) -> inf. Polynomial grows faster than any log.

Big O Notation

f=O(g): f <= c*g for large x. f=Theta(g): same growth. f=o(g): f/g -> 0 (f dominated).

Teaching Example: Compare 2x^2 vs 3x^4. Ratio = (2x^2)/(3x^4) = 2/(3x^2) -> 0. So 3x^4 dominates. In Big O: 2x^2 = O(x^4) and 3x^4 = Omega(x^2).

Applications

Algorithm Analysis Complexity Theory Data Science Physics Economics

Frequently Asked Questions

What is growth rate?▼

How fast function increases as x->inf. Hierarchy: log < poly < exp < factorial. Measured by limit of ratio.

Compare two polynomials?▼

Higher degree dominates. Same degree: compare leading coefficients. x^4 dominates x^2 regardless of coefficients.

Exp vs poly winner?▼

Exponential always wins. 2^x / x^1000000 -> inf. Exponential dominates any polynomial, no matter the degree.

Big O meaning?▼

f=O(g): f <= c*g for large x (upper bound). f=o(g): f/g -> 0. f=Theta(g): same growth rate.

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