Bendixson was born on 1 August 1861 at Villa Bergshyddan, Djurgården, Oscar Parish, Stockholm, Sweden,[1] to a middle-class family.

On June 16, 1905, he assumed the position of professor of higher mathematical analysis at Stockholm University College, and he served as its rector from 1911 until 1927.

He was well known for his mild left-wing views and he put his beliefs into practice being head of a committee to help poor students.

His first research work was on set theory and the foundations of mathematics, following the ideas that Georg Cantor had introduced.

The Poincaré–Bendixson theorem, which says an integral curve that does not end in a singular point has a limit cycle, was first proved by Henri Poincaré, but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901.