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 ∈ N, the function (x − a)nsin(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 x → 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 limx→a g(x) doesn’t exist either.
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