A.B. (a) limx→0 cos (1/x) (b) lim x→0 x cos (1/x)C.

Show that if f : (a,∞) → R is a function such that limx→∞ xf(x) = L, where L is a real number, then limx→∞ f(x) = 0. Show the above result directly. Do not simply quote a theorem.

Determine the value of each of the following limits or show that the limit does not exist.

Let f(x) = ( x if x is rational and 0 if x is irrational) . Show that f(x) is not bounded but that, for every c ∈ R, f(x) is bounded in a neighborhood of c.