Limes / limit
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Limes
A fortified border system, especially the former outer border of the Roman Empire in Europe and the Middle East.
Die Wanderung führte uns entlang der Überreste des römischen Limes.
The hike took us along the remains of the Roman Limes.
'Limes' is a Latin word that originally meant 'border path' or 'border'. Knowing this helps you understand both meanings: the historical border of the Roman Empire and the mathematical limit.
Der Limes war die befestigte Grenze des Römischen Reiches.
The Limes was the fortified border of the Roman Empire.
You can easily recognize the meaning of 'Limes' from context. Words like 'römisch' (Roman), 'Grenze' (border), 'Wandern' (hiking), or 'Archäologie' (archaeology) point to the historical meaning. Terms like 'Funktion' (function), 'Folge' (sequence), 'berechnen' (to calculate), or 'konvergiert' (converges) point to the mathematical meaning.
Wir haben den Verlauf des römischen Limes auf der Karte verfolgt.
We traced the course of the Roman Limes on the map.
In a historical context, 'Limes' is often used with the preposition 'entlang' (along) to describe movement or location. Verbs like 'verlaufen' (to run/course), 'sichern' (to secure), or 'erforschen' (to research/explore) are also typical.
Der Obergermanisch-Raetische Limes ist UNESCO-Weltkulturerbe.
The Upper Germanic-Rhaetian Limes is a UNESCO World Heritage Site.
In mathematics, 'Limes' is usually combined with verbs like 'berechnen' (to calculate), 'bestimmen' (to determine), or 'finden' (to find). One also says that a sequence or function 'gegen einen Limes strebt' (approaches a limit) or 'einen Limes hat' (has a limit).
Können Sie den Limes dieser Folge bestimmen?
Can you determine the limit of this sequence?
border / limit
'die Grenze' and 'der Limes' can both have to do with an end or a boundary. 'Grenze' is the normal word for a dividing line between countries or areas, and also for a limit in a figurative sense. 'Der Limes' is much more specific: in history, it especially means a fortified border of the Roman Empire. In mathematics, 'Limes' means a limit. The words can be confusing because 'Limes' is connected to a border or limit in both meanings. In everyday language, you almost always say 'Grenze'; 'Limes' fits mainly in history or mathematics.
Im Geschichtsunterricht lernen wir heute etwas über den Limes der Römer.
In history class today, we are learning about the Roman Limes.
Auf der Karte sieht man die Grenze zwischen den beiden Ländern deutlich.
On the map, you can clearly see the border between the two countries.