(√(3+2x)-(√2+1))/(x²-2) can be rewritten by multiplying top and bottom by:
√(3+2x)+(√2+1).
This changes the numerator to:
3+2x-(√2+1)²=3+2x-3-2√2=2(x-√2).
The denominator factorises: x²-2=(x-√2)(x+√2).
Both expressions have x-√2 as a common factor.
So, the original expression changes to:
2/((x+√2)(√(3+2x)+(√2+1)).
The common factor happens to take the value 0 when x=√2, therefore x→√2 will approach the above expression when x=√2.
This value is:
2/((√2+√2)(√(3+2√2)+(√2+1))=0.14645 approx.
So this is the limit as x→√2.