Alternating sequence, convergence and its limit

The question asks to verify whether it converges and what the limit is if it does

This is the sequence where n ∈ N

(n^4 − 3n^2 + 1)/(n− √[1 + 4n^8])

I began attacking this via the sandwich theorem, but then realised that the square root in the denominator means there are two answers for each value of n... therefore is it even an alternating sequence?

outside the brackets of the sequence there is an ∞ at the top and n=1 at the bottom

... does this mean I should only consider the positive values?

thanks