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.
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
'Grenze' and 'Grenzwert' both have to do with a point that should not simply be crossed, or that marks an important boundary. 'Grenze' is the more general word and fits in many situations, including everyday ones. 'Grenzwert' is more specific and more technical. It is often used with measurements, in medicine, the environment, technology, or mathematics. The words can be confused because both refer to a limit. If it is about a measured or fixed value, 'Grenzwert' is more natural; otherwise, 'Grenze' is usually enough.
Im Labor liegt der gemessene Wert noch unter dem erlaubten Grenzwert.
In the lab, the measured value is still below the permitted threshold.
Bei lauter Musik am Abend gibt es für meine Nachbarn eine klare Grenze.
With loud music in the evening, there is a clear limit for my neighbors.