at the crack [the break] of dawn
明け方に
at [on] the earliest opportunity
機会がありしだい
at the end of one's patience [endurance]
忍耐の限界に達して
at the end of one's tether
体力[気力,忍耐]が尽きて
at the end of the corridor
廊下のつきあたりに
at the end of this story
この話の結末で
at the entrance to a city [a house]
町[家]の入り口で(◆入り口を建物の一部と見るときはthe entrance of a buildingという)
at the expense of one's health
健康を犠牲にして
at the extremity of a penninsula
半島の先端
at the extremity of one's desperation
絶望の果てに