(a) Given that

lim _x→0 (3x + 1)(3x − 1)x² + 0.01 = 0.01

prove that there exists an open interval (a, b) containing 0 such that (3x + 1)(3x − 1)x² + 0.01 > 0 for all x ≠ 0 in (a, b).

(b) Given that lim _x→c g(x) = L, where L > 0, prove that there exists an open interval (a, b) containing c such that g(x) > 0 for all x ≠ c in (a, b).