Question

Find the limit: see the attachment. Please, don't throw around the word derivative, just yet.

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

lol i was writing instead of looking at what you wrote

Answer 6

now take the limit by replacing \(\Delta x\) by \(0\) and get
\[\frac{1}{2\sqrt{x}}\]

Answer 7

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

Answer 8

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

Answer 9

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