# Thread: [SOLVED] appropriate simplification when determining a limit of a rational function

1. ## [SOLVED] appropriate simplification when determining a limit of a rational function

I am getting a bit lost in the middle of finding the limit of the function below. Two questions.
A) Is the multiplication of the conjugate correct, and is the denominator properly expanded?
B) once the expansion is done, I still end up with a value of zero for limit when substituting...
I know that the answer is ½, but I am just not seeing how. Can you clarify it for me?
$\displaystyle \lim_{x\to0}\frac{\sqrt{1+h}-1}{h}$
$\displaystyle \lim_{x\to0}\frac{\sqrt{1+h}-1}{h}\times\frac{\sqrt{1+h}+1}{\sqrt{1+h}+1}$
$\displaystyle \lim_{x\to0}\frac{1+h-1}{h\sqrt{1+h}+h}$
$\displaystyle \lim_{x\to0}\frac{h}{h\sqrt{1+h}}$

2. $\displaystyle \lim_{h\to0}\frac{1+h-1}{h\sqrt{1+h}+h} =$

$\displaystyle = \lim_{h\to0}\frac{h}{h(\sqrt{1+h}+1)} =$

$\displaystyle = \lim_{h\to0}\frac{1}{\sqrt{1+h}+1} = \frac{1}{2}$

3. Thanks!
Sometimes I don't see the forest for the trees.