()える

たえる
verb (ichidan)
to cease, to die out, to run out
1. to cease, to come to an end, to be cut off
For something that was continuing to stop completely. Often used for communication, sound, supply, or a continuous state being interrupted.
連絡(れんらく)()えた。
Contact was cut off.
客足(きゃくあし)()えない人気(にんき)(てん)だ。
It's a popular shop where the flow of customers never stops.
(わら)いの()えない家庭(かてい)(きず)きたい。
I want to build a home that is always full of laughter.
Literally: a home where laughter never ceases.
2. to die out, to become extinct, to be discontinued
For a lineage, species, tradition, or practice to completely disappear, with no continuation remaining.
この技術(ぎじゅつ)後継者(こうけいしゃ)がおらず()えてしまった。
This craft died out because there were no successors.
家系(かけい)()える。
A family line dies out.
戦争(せんそう)(むら)伝統(でんとう)行事(ぎょうじ)()えてしまった。
The village's traditional events were lost due to the war.

GRAMMAR:
Ichidan verb (()える → ()えます). Intransitive. Very commonly used in the negative form ()えない (never ceases), which has a positive nuance: (わら)いが()えない (always laughing), (きゃく)()えない (always busy with customers).

COMMON COLLOCATIONS:

  • 連絡(れんらく)()える: contact is lost
  • (わら)いが()えない: laughter never stops
  • 客足(きゃくあし)()えない: customers keep coming
  • 家系(かけい)()える: a family line dies out
  • (いき)()える: to breathe one's last

SIMILAR WORDS:
The homophone ()える means "to endure" — a completely different word. 途絶(とだ)える is a near-synonym that emphasizes interruption of something expected to continue.