Asymptotic analysis

Asymptotic analysis studies how functions or sequences behave as an input approaches a limit, often by comparing their rates of growth or decay.

Limit (mathematics): A value that a function or sequence approaches as its input approaches a point or infinity. Asymptotic statements describe behavior precisely through limits.
Algorithm analysis: The study of an algorithm's resource use and performance as its input size varies. Asymptotic growth compares algorithm costs while abstracting away machine-specific constants.
Asymptotic expansion: A representation approximating a function by an ordered series of terms with a controlled limiting remainder. It gives successive levels of accuracy rather than only a growth bound.
Numerical analysis: The study of algorithms for obtaining numerical solutions to mathematical problems. It often seeks finite-input accuracy, while asymptotic analysis emphasizes limiting behavior.
Big O notation: A notation expressing an eventual upper bound on a function's magnitude up to a constant factor. It gives a standard way to bound growth while ignoring constant factors.
Time complexity: A measure of how an algorithm's running time grows with input size. Its asymptotic form classifies long-run runtime scalability.
Taylor series: A power series built from a function's derivatives at a point. Taylor polynomials yield local asymptotic approximations when smoothness permits.
Exact solution: A solution expressed without approximation and satisfying a problem's equations exactly. Asymptotic formulas approximate behavior and need not provide an exact expression.
Little o notation: A notation stating that one function becomes negligible relative to another in a specified limit. It distinguishes strict lower-order terms from terms merely bounded by a scale.
Perturbation theory: Methods for approximating solutions to problems that depend on a small parameter. Asymptotic expansions often supply approximations when exact solutions are unavailable.

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