approximation - Approximating for $a gg b$

I'd like to know how I could go about making approximations where one quantity is much smaller or larger than another.

For example, the expression $\frac{1}{b(a +b)}$ is approximately equal to $\frac{1}{ab}$ when $a \gg b$

But of course, simply taking the infinite limit of a does not yield the right result

i.e.

Limit[1/((a + b) b), a -> ∞]

gives a result of 0. It is the ratio of $a$ and $b$ that needs to approach infinity. One cannot do this directly via

Limit[1/((a + b) b), a/b -> ∞]

and one can get the right answer by substituting an explicit ratio via transformations such as /. (a -> r b)

So how do I get the result I require?

Answer

How about this:

Normal@Series[1/((a + b) b), {a, Infinity, 1}]

(* ==> 1/(a b) *)

Normal@Series[ArcTan[a + b], {a, Infinity, 1}]

(* ==> -(1/a) + Pi/2 *)

Edit in response to comment

Having been told what the desired result for ArcTan[a+b] is, it looks like the following expansion method might be what's needed. At least it's consistent with the information provided so far:

Normal[1/((a + O[b] + b) b)]

(* ==> 1/(a b) *)

Normal[ArcTan[a + b + O[b]]]

(* ==> ArcTan[a] *)

This uses the big-o notation, O, directly in the expression.

To automate what I did above, I would use the following Rule to implement the statement $b\ll a$ in a given expression:

expression/.a->(a+O[b])

This will cause any powers $b^n$ with $n> 0$ to be dropped when they appear in a sum with $a$.