# Thread: Seeking limit of function with expression radicand in denominator

1. ## Seeking limit of function with expression radicand in denominator

Hi all,

I am in an early calculus chapter, where I am finding derivatives algebraically. (Hence why I am posting in pre-calculus; I have not yet covered techniques to differentiate functions efficiently.) I am asked to find the differential of the following function:

${f}(x) = \frac{1}{\sqrt{x+2}}$

I am trying to find the derivative by using a limit function to determine instantaneous rate of change:

${f}'(x) = \lim_{h \to 0} \left ( \frac{1}{h} \right ) \left (\frac{1}{\sqrt{x + h + 2}} - \frac{1}{\sqrt{x + 2}} \right )$

However, I cannot solve this equation--I don't know how to get an 'h' factor in the numerator, which would allow me to cancel out the (1 / h) term. I have tried multiplying, dividing, and subtracting terms; I have tried multiplying the conjugates of radicals; I've been at this one for a while. Hopefully, one of you thinks this is obvious, and I've already over-explained a very simple problem... I've been staring at this one too hard, and I'm definitely missing the trick.

Thank you!

2. ## Re: Seeking limit of function with expression radicand in denominator

Get a common denominator to combine the two terms. This gives:

$\displaystyle \lim_{h\to 0} \dfrac{\sqrt{x+2}-\sqrt{x+h+2}}{h\sqrt{(x+h+2)(x+2)}}$

Then multiply top and bottom by the conjugate.

$\displaystyle \lim_{h\to 0}\dfrac{-h}{h\sqrt{(x+h+2)(x+2)}\left(\sqrt{x+2}+ \sqrt{x+h+2}\right)}$

Can you solve it from there?

3. ## Re: Seeking limit of function with expression radicand in denominator

It looks so obvious when you do it. Thank you!