I'm not sure what your question is or exactly HOW you "rationalized". The point of rationalizing is that (a- b)(a+ b)= a^2- b^2 so if a or b involved a square root, we will have gotten rid of it by squaring.
I could see two ways of approaching .
a) Multiply by to get
so the numerator can be written as and then, divided by h,
so that, with the denominator, we have
The limit, as h goes to 0 is
Personally, I would combine the fractions first:
Now, multiply numerator and denominator by to get
Dividing by h and taking the limit as h goes to 0 gives the same as before.