Name :  Pooja  Sharma

Title : "The Growth of Functions"

E -mail : psharma105@qc.cuny.edu

s[ax_,bx_]:=If[Limit[ax/bx,x->∞]==0,  "is little oh of" , "is little omega of"];
(* This function takes two variables and check them as the limit goes to infinity. If the function goes to zero then ax is little oh of bx, otherwise ax is little omega of bx.*)

Manipulate[{ax,s[ax,bx],bx},(*Manipulate command will print out the results accordingly to the 's' function.*){ax,{x,x^2,x!,x/Log[x],x^1000,1,x!,Log[x]!},ControlType->PopupMenu},(*ControlType makes the 'ax' as a popup menu and in 'ax' menu there are options such as x, x/Log(x) etc.*){bx,{Log[x], x^3,2^x,1/x,(Sqrt[x]),x Log[x],(2+x)!},ControlType->PopupMenu},
(*This command is same as above; With the help of controlType I make 'bx' as pop up menu and insert some other functions such as x Log(x), 2^x, etc.*)SaveDefinitions->True](* SaveDefinitions is an option to manipulate and related functions which specifies whether current definitions relevant for the evaluation of the expression being manipulated should automatically be saved .*)

Just for fun :-  To give you the more clear idea that how little oh and little omega varies with different functions

I made up a small table using "Grid" command and in this table f(x) is little oh of g(x) or in other words, g(x) is little omega of f(x).

Grid[ (*Grid is an oject that makes the two dimensional format. *){{Item[f[x],Background->LightPurple],Item[g[x],Background->LightBlue]}, (*Item is a function which makes f(x) and g(x) as the main title for two columns*){ x, (x^2)}, { Log[x],x/Log[x]},

(*And fill the data row wise below f(x) and g(x)*){(n^x),(x^n)},{ 1, Log[x]},{x, x Log[x]},{ x!,(x+2)!},{(x^2),.0001(x^3)},{Log[x],(Sqrt[x])}},ItemSize->{6,2}(*ItemSize makes each item a fixed number of character-widths wide*),Frame->All]