That depends whether we're talking about on-chain/off-chain and use SegWit/not. If we're only talking about tx/s on-chain, then it would be :
1. about 7 tx/s if all transaction don't use SegWit
2. between 12-20 tx/s if all transaction use SegWit
But if we're talking about off-chain tx/s, then in theory it's unlimited (ignoring on-chain transaction to open and close channel)
I think the unlimited theory is true and will stand true, but also, not only ignoring the on-chain requirement to open/close channel and settle final books, the amount in the channel will somewhat limit the #s of simultaneous transactions... for example, if the channel only has say 1 million satoshi, presumably it can handle a max of 1 million txs each of 1 satoshi going in one way.
Practically also, LN channels have trouble with higher amounts, but this exact amount is increasing daily.