Hello, I am assuming that you want the limit as n->Infinity? If so,
expr = (2 n + 1)/(n - 2)
eqn = expr - Limit[expr, n -> Infinity] == epsilon
sol = Solve[eqn, n]
n /. sol /. epsilon -> 0.01
gives n approximately equal to 502.
Plot[Evaluate[expr - Limit[expr, n -> Infinity]], {n, 490, 510},
PlotRange -> All, Epilog -> Line[{{0, .01}, {1000, .01}}]]
