One of the nice things about big-O notation from a mathematical point of view is that it allows you to avoid dealing with limits. As everyone knows, calculus i

One of the nice things about big-O notation from a mathematical point of view is that it allows you to avoid dealing with limits. As everyone knows, calculus is based on limits, but when handled rigorously limits have a lot of subtle pitfalls that take a lot of work to deal with. For example, given a function f(x,y), if we want to take one limit after another, the result might depend on the order in which the limits are taken. In other words, taking limits is not commutative unless certain conditions are satisfied. None other than the great Donald Knuth went so far as to claim that all of calculus could be taught without limits! Quoting from [1]: "Students will be motivated to use O notation for two important reasons. First, it significantly simplifies calculations because it allows us to be sloppy — but in a satisfactorily controlled way. Second, it appears in the power series calculations of symbolic algebra systems like Maple and Mathematica, which today’s students will surely be using. For more than 20 years I have dreamed of writing a calculus text entitled O Calculus, in which the subject would be taught along the lines sketched above." [1] http://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/ http://micromath.wordpress.com/2008/04/14/donald-knuth-calcu...