I need to find the derivative of $\displaystyle f(x)=x+\sqrt x+3$

I have gotten this far: $\displaystyle f'(x)=lim_{h\to0}\frac{f(x+h)-f(x)}{h}=lim_{h\to0}\frac{(x+h)+\sqrt{x+h} +3-\left(x+\sqrt x +3\right)}{h}=lim_{h\to0}\frac{h+\sqrt{x+h}-\sqrt x}{h}$

At this point I have tried some crazy stuff to eliminate the h on the bottom. The closest I have come is this: $\displaystyle lim_{h\to0}\frac{h}{h}+\frac{\sqrt{x+h}}{h}-\frac{\sqrt x}{h}$ which can be broken down into, $\displaystyle lim_{h\to0}\frac{h}{h}+\frac{\sqrt x}{h}+\frac{\sqrt h}{h}-\frac{\sqrt x}{h}$ which leaves this:$\displaystyle lim_{h\to0}\frac{h}{h}+\frac{\sqrt h}{h}$, right?

If I haven't confused myself along the way, then I should be on track, and the answer seems tantilizingly close, but I get $\displaystyle lim_{h\to0}1+\frac{\sqrt h}{h}$=$\displaystyle lim_{h\to0}1+\frac{1}{\sqrt h}$

What am I doing wrong?