In your second approach, you actually made the same error that you made in your first approach. I am with you up to this point:

Now, when you took the limit as

you used a denominator of 6 for the last term within the parentheses, when it should be a 3, do you see why?

I will let you rework this on your own once you understand why you need a 3 there. Everything else you did is correct except for that.

edit: If you have a bunch of quadratic functions, you may want to generalize further and use

on

.