Question 1044609
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right p}\left\[\frac{\sqrt{x\,-\,q}\ -\ \sqrt{p\,-\,q}}{x^2\ -\ p^2}\right\]]


Rationalize the numerator:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right p}\left[\left(\frac{\sqrt{x\,-\,q}\ -\ \sqrt{p\,-\,q}}{x^2\ -\ p^2}\right)\left(\frac{\sqrt{x\,-\,q}\ +\ \sqrt{p\,-\,q}}{\sqrt{x\,-\,q}\ +\ \sqrt{p\,-\,q}}\right)\right\]]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right p}\left[\frac{x\,-\,q\ -\ (p\,-\,q)}{\left(x^2\ -\ p^2\right)\left(\sqrt{x\,-\,q}\ +\ \sqrt{p\,-\,q}\right)\right\]]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right p}\left[\frac{x\,-\,p}{\left(x\ -\ p\right)\left(x\ +\ p\right)\left(\sqrt{x\,-\,q}\ +\ \sqrt{p\,-\,q}\right)\right\]]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right p}\left[\frac{1}{\left(x\ +\ p\right)\left(\sqrt{x\,-\,q}\ +\ \sqrt{p\,-\,q}\right)\right\]]


Evaluate at *[tex \Large x\ =\ p]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right p}\left[\frac{1}{\left(x\ +\ p\right)\left(\sqrt{x\,-\,q}\ +\ \sqrt{p\,-\,q}\right)\right\]\ =\ \frac{1}{2p\left(2\sqrt{p\,-\,q}\right)}\ =\ \frac{1}{4p\left(\sqrt{p\,-\,q}\right)]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  

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