(Requires calculus) The two parts of this exercise describe the relationship between little-o and big-O notation.

a) Show that if f(x) and, g(x) are functions such that f(x) is o(g(x)), then f(x) is O(g(x)).

b) Show that if f(x) and g(x) are functions such that f(x) is O(g(x)), then it does not necessarily follow that f(x) is o(g(x)).

Step 1:

In this problem, we have to show that the relation between big-O and little-o for the function.

Step 2:

The definition for Big- O:

Let f and g be functions from the real numbers to the real numbers. Then f is O(g) if there are constants c and k

Such that

That means O(g(x)) is the set of all functions with a smaller or the same order of growth as f(x).

Example: O(x2) = {x2 , 50x +40 , logx,.....}

The definition for little-o:

Let f and g are two functions, and o(g(x)) is the set of all functions with a smaller rate of growth than f(x).

Example: o(x2) = {50x +40, logx,...}