The definition of velocity I learned from a book that Newton invent the calculus with the concept of velocity . However, the definition of velocity needs the concept of limit, which was not invented at that time. How did they define velocity ?

Newton certainly was familiar with the concept of limits, although they were not called by that name and he almost certainly did not apply the rigor of the $\delta, \epsilon$ definition, which was developed by Weirstrass (I probably spelled that wrong) surprisingly recently (about 1900, I think).

Leibnitz also worked with limits and infinitessimals, although again he did not use the modern real analysis approach in defining them.