Hi I'm just wondering what steps Evans takes in his PDE book on page 50.

He defines

And then goes on to say:

.

I do not know how he got here:

I think I can get this far:

Taking the limit as , I can kind of see how the left term on the RHS can go to , but even this I'm not sure of, I don't really understand, I can't really see why it shouldn't it be since isn't this the 'variables' that is having an infinitesimally small h added on to? I have no idea how the right term of the RHS of the above equation transforms as well. I think the thing that is most troubling me is that the variable t is in the integral and the integrand and what we are taking a limit of, h, is in the integral... and in the integrand once we take the 1/h back inside!

Can someone please help?! Thanks!