## MoonlitFate 2 years ago Find the limit: see the attachment. Please, don't throw around the word derivative, just yet.

1. MoonlitFate

2. MoonlitFate

I can do the first step, which I think is to rationalize the numerator, but then I get stuck.

3. satellite73

rationalize the numerator by multiplying top and bottom by the conjugate of the numerator when you do, the $$\Delta x$$ will cancel

4. MoonlitFate

Then I get: $\frac{ 1 }{ \sqrt{x + \Delta x } + \sqrt{x}}$

5. satellite73

$\lim_{h\to 0}\frac{\sqrt{x+h}-\sqrt{x}}{h}\times \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$ $=\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}$ $=\frac{h}{\sqrt{x+h}+\sqrt{x}}$ $=\frac{1}{\sqrt{x+h}+\sqrt{x}}$

6. satellite73

lol i was writing instead of looking at what you wrote

7. satellite73

now take the limit by replacing $$\Delta x$$ by $$0$$ and get $\frac{1}{2\sqrt{x}}$

8. satellite73

which happens to be the "DERIVATIVE" (yeah, i know) of $$\sqrt{x}$$ a good one to memorize

9. MoonlitFate

Wow. Funny how the small steps confuse me, yet, I can do most of the "hard" things...

10. satellite73

true, you did all the hard work replacing $$\Delta x$$ by zero is rather simple