limitsroughly

Limitsroughly is a conceptual term used to describe an approximate limit in mathematical and numerical contexts. It denotes the idea that a sequence or a function approaches a target value within a specified tolerance rather than converging exactly.

For a sequence a_n, we say it has a limit roughly L with tolerance ε > 0 if there exists N such that for all n ≥ N, |a_n - L| ≤ ε. For a function f with a limit at a, we say it has a limit roughly L as x → a if there exists δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| ≤ ε. In both cases, the concept depends on a chosen ε, which fixes the closeness requirement.

This contrasts with the standard notion of limit, which requires that for every ε > 0 there is a corresponding N or δ. Limitsroughly is therefore a pragmatic, tolerance-based view useful in computation, data analysis, and teaching when exact limits are unattainable or unnecessary.

Example: a_n = 2 + 1/n has a limit roughly 2 with ε = 0.01, occurring for n ≥ 100. A_n = sin(n) does not have a limit, though one can discuss a rough limit within a chosen ε over a bounded index range, or in a probabilistic sense.

Use and limitations: Limitsroughly provides a concise language for describing approximations but does not replace formal convergence in proofs. It is appropriate when error bounds and numerical tolerances matter, but should be complemented by standard convergence concepts in rigorous analysis. Related topics include convergence, error tolerance, and numerical analysis.