Born in Western Australia, he got a Bachelor of Electrical Engineering (B.E.E.) (1963) and a Ph.D. on the thesis Vision Oriented Coding of Visual Signal (1967), both from the University of Western Australia. Limb studied the human vision's role in the encoding of picture, partially under the direction of Dr. Albert Seyler and Professor Zig Budrikis.
He worked as researcher at the Postmaster-General's Department's research laboratory in Melbourne (1966–67) before moving to New Jersey where he joined Bell Labs (1967), being manager of the department for visual communications research (1971–78) leading to patents. He then led the research department for distributed computer systems (1978–84) before going to Bellcore research where Limb was in charge of developing the new research division for communications sciences (1984–86) where he among other projects, developed medium access control protocols for local area networks. Later he joined Hewlett Packard as director of laboratories in Bristol, England (1986–89), for the technology analysis group at Cupertino, California (1989) and for the media technology group (1992–94). Distributed Computer Systems Research Department where the emphasis was on office information systems and local/metropolitan area networks. Recently, he has worked on optical communication systems at Broadcom, working at Atlanta, Georgia.
Limb started at the Georgia Institute of Technology in 1994 as a Georgia Research Alliance Eminent Scholar in Advanced Telecommunications. He joined the faculty staff of both the Networking & Telecommunications Group in the College of Computing, and the School of Electrical and Computer Engineering in the College of Engineering. He created the Georgia Tech Broadband Telecommunications Center in 1995.