$\displaystyle \lim_{x \to 1} x^\frac{1}{(1-x)}$
Answer:
-1
Hi
$\displaystyle x^{\frac{1}{1-x}}=\exp\left( \frac{\ln x}{1-x}\right)$
$\displaystyle \ln x\underset{x\to1}{\to}0$ and $\displaystyle 1-x \underset{x\to1}{\to}0$ hence one can use l'Hospital's rule : $\displaystyle \lim_{x\to1}\frac{\ln x}{1-x}=\lim_{x\to1}\frac{\frac{1}{x}}{-1}=-1$ thus $\displaystyle x^{\frac{1}{1-x}}\underset{x\to1}{\to}\frac{1}{\mathrm{e}}$
technically they are the same thing. it should really be pronounced Lopital (the s is silent), but there are two different spellings to it. the original spelling is L'Hospital, but if you write it as L'hopital, technically there should something over the o to show that the s was omitted, so you should write $\displaystyle l'H\hat opital$ .....i think capitalizing the L in front as opposed to the H is also optional. yes, it's weird.
i forgot what that thing over the o is called. it's not a tilde. the LaTeX command is \hat, but i doubt it's called a hat...
The reality of the difference between L'Hospital and L'Hôpital : The man we call " L'Hospital", or "L'Hôpital" was in fact L'Hospital. (Which was the French word for "The Hospital" at this time. But since this word evoluted into Hôpital (in French of course. And many other words containing an s from Latin has been "cut" and to know there was an s, we put a circumflex accent ^ (in other case an acute accent)). For example, the word Insula from Latin (island in English) became île (when it's still Isla in Spanish for example), we use to call him nowadays L'Hôpital.
True...just like Taylor and Maclaurin Series...
Taylor didn't invent Taylor Series...James Gregory was working with Taylor Series when Taylor was only a few years old, and he had developed the Maclaurin series for tan(x), sec(x), arctan(x), and arcsec(x) 10 years before Maclaurin was born...
Taylor published his book on series [not knowing about this previous work] and this is where the series became known as "Taylor" Series...
It is also interesting to note that Maclaurin discovered Cramer's Rule...
This is some interesting stuff.
--Chris