I am studying the use of Big O notation online ( use in Taylor Series Etc.) but I am not understanding it right from its definition itself. It says, f is said to be in O(x^(n)) if |f(x)|0 and x->infinity. After that I have been given following examples, Cases as x->0 1. 5x+3x^(2) is in O(x) but not in O(x^(2)) 2. Sin(x) is in O(x) but not in O(x^(2)). 3. ln(1+x)-x is in O(x^(2)) but not in O(x^(3)). 4. 1-cos(x^(2)) is in O(x^(4)) but not in O(x^(5)). 5. sqrt(x) is not in O(x^(n)) for any n>=1. 6. e^(-1/x^(2)) is in O(x^(n)) for all n. Cases as x-> infinity 1. arctan(x) is in O(1) as well as O(x^(n)) for any n>=0. 2. x*sqrt(1+x^(2)) is in O(x^(2) but not in O(x^(3/2)). and many more. I am not able to apply the definition to all above examples to check. Can you please help. I am familier with the relative growth of functions, Loptials rule etc. Thanks and regards, Rahul.