# Thread: Limit Defintion of the Derivative/ Rationalization help

1. ## Limit Defintion of the Derivative/ Rationalization help

I have the function f(x)= the square root of(x+2). i must find the limit of x=2 as h approaches zero. When i rationalize the numerator in the problem i get (4h-4)/(2h times the square root of (h) +2h). I do not see anyway to simplify the problem. Can anyone see what im doing wrong/ not doing?

2. Originally Posted by gamefreeze
I have the function f(x)= the square root of(x+2). i must find the limit of x=2 as h approaches zero. When i rationalize the numerator in the problem i get (4h-4)/(2h times the square root of (h) +2h). I do not see anyway to simplify the problem. Can anyone see what im doing wrong/ not doing?
$\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}=$

$\lim_{h \to 0}\frac{\sqrt{x+h+2}-\sqrt{x+h}}{h}=$

Now you want to multiply by the conjugate of the numerator to get

$\lim_{h \to 0}\frac{\sqrt{x+h+2}-\sqrt{x+h}}{h}\cdot \frac{\sqrt{x+h+2}+\sqrt{x+2}}{\sqrt{x+h+2}+\sqrt{ x+2}}=$

$\lim_{h \to 0}\frac{h}{h(\sqrt{x+h+2}+\sqrt{x+2})}=\lim_{h \to 0}\frac{1}{\sqrt{x+h+2}+\sqrt{x+2}}=\frac{1}{2\sqr t{x+2}}$

3. thank you very much. So it is easier to not input the values of x until i am done simplifying?

4. Originally Posted by gamefreeze
thank you very much. So it is easier to not input the values of x until i am done simplifying?
It is a matter of opinion. I personally think so, but it can go either way. The reason I like to do it last is because it is more general, the formula is valid for any x, not just x=2.

5. ok thanks a lot
when not plugging in the values of x, the algebra is much easier to do.