Please describe (specifically) how to solve the problem posted here:

http://alethiography.blogspot.com/2006/05/calculus-problem-1-limit-challenge.html

In other words, I want the steps to solve this problem:
 
find the limit, as h approaches 0, of

(sqrt(1 + 2(x+h)) - sqrt(1 + 2x)) / h

I know the correct answer, so what I am looking for is the actual
steps taken to get there - the algebra needed to make the formula
above (or posted much more neatly at the link above) solvable when h =
0.

Just to be clear, I do not want just the answer, nor do I want an
alternative-math way of solving this (for instance, by using the chain
rule to attain the derivative of sqrt(1 + 2x)) - what I'm trying to
learn is the underlying technique needed to solve this type of
problem.

Hint: Multiply both the numerator and denominator by [sqrt(1 + 2(x+h)
) + sqrt(1+2x) ].

Thank you!  this works :-)