Please help me with this prove question

Prove that if there is a number B such that |f(x)| <= B for all x not = 0, then lim x approaches 0 xf(x) = 0. Note, exercise 43-46 are special cases of this gerenal result.

Special cases are:

43. lim x approaches 0 sin (1/x) = 0;

44. lim x approaches pi (x - pi) cos^2 [1/(x-pi)] = 0;

45. lim x approaches 1 |x -1| sinx = 0;

46.

f(x) = {1 if x rational

f(x) = {0, if x irrational then lim x approaches 0 xf(x) = 0.

**This is the hardest question in my assgnment, no one in my study group is able to do it, I really need anyone who can give me some hints on this, thanks!**