threshold / limit
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threshold
A fixed maximum or minimum value that must not be exceeded or fallen below in a particular area (e.g., environment, technology, medicine).
Die Messungen ergaben, dass der Grenzwert für Stickoxide in der Innenstadt überschritten wurde.
The measurements showed that the threshold for nitrogen oxides in the city center had been exceeded.
limit
In mathematics, the value that a function or a sequence approaches. A synonym for limit.
Der Grenzwert der Funktion für x gegen unendlich ist null.
The limit of the function as x approaches infinity is zero.
The word is a clear compound of 'Grenze' (border/limit) and 'Wert' (value). So it literally refers to a value that represents a limit and should not be exceeded or fallen below.
Für den Lärm in Wohngebieten gibt es klare Grenzwerte.
There are clear thresholds for noise in residential areas.
In general usage, 'Grenzwert' (threshold/limit) is often used with certain verbs. Typical phrases are 'einen Grenzwert überschreiten' (to exceed a threshold), 'einen Grenzwert einhalten' (to comply with a threshold), or 'einen Grenzwert festlegen' (to set a threshold).
Die Fabrik muss sicherstellen, dass sie die gesetzlichen Grenzwerte einhält.
The factory must ensure that it complies with the legal thresholds.
The meaning as a mathematical limit is purely technical language. Outside of math class or science, you will hardly encounter this usage.
Der Professor erklärte, wie man den Grenzwert einer Funktion bestimmt.
The professor explained how to determine the limit of a function.
In everyday life and in the news, 'Grenzwert' (threshold/limit) almost always refers to environmental, medical, or technical topics. In a mathematical context, it means the limit of a function or sequence.
Die Wasserqualität ist gut, alle Grenzwerte werden eingehalten.
The water quality is good; all thresholds are being met.
border / limit
Die Grenze and der Grenzwert can both refer to an endpoint or a limit. In mathematics, however, der Grenzwert is the exact value that a sequence or function approaches, while Grenze remains more general. When talking about this mathematical approach, Grenzwert is more appropriate.
Der Grenzwert der Folge ist zwei.
The limit of the sequence is two.
Diese Grenze darf die Funktion nicht überschreiten.
The function must not exceed this limit.
