végértékhez

Végértékhez is a term used in Hungarian mathematics to express motion toward the limit value of a sequence or a function. The végérték, or end value, is the value that the values of a sequence or the values of a function approach as the index tends to infinity or as the argument tends to a specified point. The form végértékhez (to the limit value) is a directional phrase used with verbs such as közelít (approaches), tart (approaches and stays near), vagy megy (goes toward).

In the context of sequences, if a_n is a sequence in a metric space and lim_{n→∞} a_n = L, then L is the végérték, and one can say that a_n közelít a végértékhez, i.e., a_n approaches the limit value L. In function theory, if lim_{x→a} f(x) = L, then L is the végérték of f at a, and as x approaches a, f(x) tends toward the végértékhez.

Notationally, the concept is equivalent to the more common term határérték (limit). The distinction lies mainly in terminology rather than in meaning, and végértékhez is frequently found in descriptive or problem-solving contexts.

Common methods to determine végerértékek include standard limit techniques: algebraic simplification, factoring, rationalizing, common limits rules, the squeeze theorem, and, for more complex cases, L’Hôpital’s rule or series expansions.

Examples:

- a_n = 1/n has végerérték = 0; a_n közelít a végértékhez 0.

- For x ≠ 1, f(x) = (x^2 − 1)/(x − 1) simplifies to x + 1, and as x → 1, f(x) → 2, so végértékhez 2 tart.

The concept is foundational in analysis, linking convergence to a concrete target value and guiding the assessment of stability and limits in mathematical models.