I am not sure this is fully rigorous: for instance, don't you need to prove first that the limit exists? (and a few steps would need justifications, like or the previous one)
I would rather say: readily, , hence . Since (and is continuous at 0), the limit indeed exists and equals 1.
I was actually a bit euphemistic in my answer : assuming the existence of the limit in the first place and playing with it is really really not a good habit to take (no harm intended!). Especially when, like here, we are asked "if the limit exists, establish the convergence"; but certainly not only in this case.
It is not because the only "possible" limit is 1 that the sequence converges to 1.
I'm just saying...