Along one specific route to the singularity, we may reach a true singular point or asymptote.

Think of computers becoming twice as fast (per dollar) every second year (roughly Moore's law). Now imagine that we have solved the problem of machine intelligence (by developing true AI or copying human intelligence to a substrate that's subject to Moore's law).

The intelligent machines will build faster computers at the same speed as humans, meaning in another two years they'll have twice the speed. The new computers become hosts to the machine intelligence and, of course, start building faster computers. Since they are twice as fast, they'll double the speed in one year. The new machines will build machines that are, again, twice as fast in six months, then three months, 1.5 months and so on.

Within four years of reaching human-level machine intelligence we'll (asymptotically) reach infinite computational power (and intelligence?), assuming there are no fundamental physical limits to computation that cannot be circumvented (a big assumption). That would be a true singularity, worthy of the name.