In 1960 Abraham Robinson invented the hyperreal number system, which contains infinitely small and large numbers. They behave as you would expect. The branch of

In 1960 Abraham Robinson invented the hyperreal number system, which contains infinitely small and large numbers. They behave as you would expect. The branch of analysis (including calculus) based on the hyperreals is called "nonstandard analysis". So you can develop calculus without the limit concept using this number system.