Let a ∊ R and let f and g be real functions defined at all points x in some open interval containing a except possibly at x = a. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples for the false ones.
a) For each n e N, the function (x - a)n sin(f(x)(x - a)-n) has a limit as x → a.
b) Suppose that {xn} is a sequence converging to a with xn ≠ a. If f(xn) → L as n → ∞, then f(x) → L as x → a.
c) If f and g are finite valued on the open interval (a - 1, a + 1) and f(x) → 0 as x → a, then f(x)g(x) → 0 as → a.
d) If limx→a f(x) does not exist and f(x) < g(x) for all x in some open interval I containing a, then lirnx→a g(x) doesn't exist either.