Used to single out X (often with a negative or unexpected result): 'of all times/people/things, especially X (and unfortunately...)'. It can also mean 'only X' in contexts that stress exception or exclusivity.
忙しい日に限って、急な会議が入った。
Of all days, a sudden meeting was scheduled on the busy day.
母に限ってそんな嘘をつくはずがない。
Of all people, my mother would never tell a lie (used to defend/exonerate).
雨の日に限って傘を忘れてしまうことが多い。
It often happens that I, of all days, forget my umbrella on a rainy day.
普段は問題ないパソコンが、テストに限って動作がおかしくなる。
A computer that normally works fine, of all times, malfunctions during the test.