Limiettheorema

Limiettheorema is a Dutch term referring to the collection of theorems about limits in calculus and real analysis. Theorems in this area describe how limits behave under basic operations, compositions, and in different mathematical contexts, such as sequences, functions, and metric spaces. They provide the rules that allow one to compute or deduce limits of more complex expressions from the limits of simpler ones.

For sequences, common limit results include the linearity of limits: if a_n tends to A and b_n tends to B, then a_n + b_n tends to A + B, a_n − b_n tends to A − B, and a_n b_n tends to AB. If B is nonzero, a_n / b_n tends to A / B. A monotone and bounded sequence converges, by the basic convergence principle for real sequences.

For functions, limit laws say that the limit of a sum or product is the sum or product of the limits, provided those limits exist. If f(x) tends to L as x approaches a, and g is continuous at L, then g(f(x)) tends to g(L) as x approaches a. The Squeeze Theorem gives a way to establish a limit when a function is trapped between two others with the same limit. These results also underpin the concept that limits can pass through compositions under appropriate continuity assumptions.

In practice, limiettheorema underpins much of calculus and analysis, enabling rigorous justification for passing to limits in algebraic manipulations, series, and integral calculations. The exact statements vary by context (real numbers, sequences, functions, or more general spaces), but the overarching idea is to formalize how limits behave under standard operations.