1. ## L'Hospital

$\displaystyle \lim_{x \to 1} x^\frac{1}{(1-x)}$

-1

2. Hi
Originally Posted by Apprentice123
$\displaystyle \lim_{x \to 1} x^\frac{1}{(1-x)}$

-1
$\displaystyle x^{\frac{1}{1-x}}=\exp\left( \frac{\ln x}{1-x}\right)$

$\displaystyle \ln x\underset{x\to1}{\to}0$ 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 $\displaystyle x^{\frac{1}{1-x}}\underset{x\to1}{\to}\frac{1}{\mathrm{e}}$

3. i was just wondering, but is it common for american mathematicians to refer to this rule as l'hospital's rule? and is this term more commonly used than the term "L'hopital's rule"?

4. Originally Posted by squarerootof2
i was just wondering, but is it common for american mathematicians to refer to this rule as l'hospital's rule? and is this term more commonly used than the term "L'hopital's rule"?
50-50...I say L'hopital

5. Originally Posted by squarerootof2
i was just wondering, but is it common for american mathematicians to refer to this rule as l'hospital's rule? and is this term more commonly used than the term "L'hopital's rule"?
Originally Posted by Mathstud28
50-50...I say L'hopital
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...

6. Originally Posted by Jhevon
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...
Accent circum-flex

7. Originally Posted by Jhevon
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...

Circumflex

8. 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.

9. to my knowledge this rule is named after the french mathematician who founded it, whose name was L'hopital. i was just wondering if his name was translated into conventional english =) anyways, thanks for the inputs.

10. Originally Posted by squarerootof2
to my knowledge this rule is named after the french mathematician who founded it, whose name was L'hopital. i was just wondering if his name was translated into conventional english =) anyways, thanks for the inputs.
He did not invent it, he bought it off of another mathematician.

11. Originally Posted by Mathstud28
He did not invent it, he bought it off of another mathematician.
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

12. Originally Posted by Chris L T521
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
Are you sure Maclaurin discovered Cramer's rule...I thought it was some monk...

13. Originally Posted by flyingsquirrel
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 $\displaystyle x^{\frac{1}{1-x}}\underset{x\to1}{\to}\frac{1}{\mathrm{e}}$

$\displaystyle ln \lim_{x \to 1} y= \lim_{x \to 1} -1$

As was $\displaystyle \frac{1}{e}$?

14. Originally Posted by Apprentice123
$\displaystyle ln \lim_{x \to 1} y= \lim_{x \to 1} -1$

As was $\displaystyle \frac{1}{e}$?
YEs

$\displaystyle \ln(y)=-1\Rightarrow{y=\frac{1}{e}}$

15. Originally Posted by Mathstud28
YEs

$\displaystyle \ln(y)=-1\Rightarrow{y=\frac{1}{e}}$

Can you explain why?

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