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

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I can do the first step, which I think is to rationalize the numerator, but then I get stuck.
rationalize the numerator by multiplying top and bottom by the conjugate of the numerator when you do, the \(\Delta x\) will cancel

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Then I get: \[\frac{ 1 }{ \sqrt{x + \Delta x } + \sqrt{x}} \]
\[\lim_{h\to 0}\frac{\sqrt{x+h}-\sqrt{x}}{h}\times \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\] \[=\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}\] \[=\frac{h}{\sqrt{x+h}+\sqrt{x}}\] \[=\frac{1}{\sqrt{x+h}+\sqrt{x}}\]
lol i was writing instead of looking at what you wrote
now take the limit by replacing \(\Delta x\) by \(0\) and get \[\frac{1}{2\sqrt{x}}\]
which happens to be the "DERIVATIVE" (yeah, i know) of \(\sqrt{x}\) a good one to memorize
Wow. Funny how the small steps confuse me, yet, I can do most of the "hard" things...
true, you did all the hard work replacing \(\Delta x\) by zero is rather simple

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