# About the real utility of l'Hôpital's rule

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• April 12th 2008, 06:56 AM
Moo
About the real utility of l'Hôpital's rule
Hello,

Looking at which frequency l'Hôpital's rule is used, i'd like to know what is its real utility... D'you have examples where we HAVE to use it ? And can't use anything else ?

As a memento :

If $\lim_{x \to a} f(x)=\lim_{x \to a} g(x)=0 \text{\ or \ } \infty$, then :

$\lim_{x \to a} \frac{f(x)}{g(x)}=\lim_{x \to a} \frac{f'(x)}{g'(x)}$
• April 12th 2008, 07:00 AM
angel.white
Quote:

Originally Posted by Moo
Hello,

Looking at which frequency l'Hôpital's rule is used, i'd like to know what is its real utility... D'you have examples where we HAVE to use it ? And can't use anything else ?

As a memento :

If $\lim_{x \to a} f(x)=\lim_{x \to a} g(x)=0 \text{\ or \ } \infty$, then :

$\lim_{x \to a} \frac{f(x)}{g(x)}=\lim_{x \to a} \frac{f'(x)}{g'(x)}$

I'd call it a useful tool, but also a potential crutch. Something which should be used when it is the most viable solution, but should be considered after other methods, lest certain skills atrophy.
• April 12th 2008, 07:37 AM
Moo
Quote:

Originally Posted by angel.white
I'd call it a useful tool, but also a potential crutch. Something which should be used when it is the most viable solution, but should be considered after other methods, lest certain skills atrophy.

This is quite the idea i had, but i only knew about how it is learnt in France (understand that it's not used wildly even never used) and a little from the USA when i read a remark from one of you.
• April 12th 2008, 07:43 AM
Mathstud28
Ok
Quote:

Originally Posted by Moo
Hello,

Looking at which frequency l'Hôpital's rule is used, i'd like to know what is its real utility... D'you have examples where we HAVE to use it ? And can't use anything else ?

As a memento :

If $\lim_{x \to a} f(x)=\lim_{x \to a} g(x)=0 \text{\ or \ } \infty$, then :

$\lim_{x \to a} \frac{f(x)}{g(x)}=\lim_{x \to a} \frac{f'(x)}{g'(x)}$

the best example would be something like $lim_{x \to 0}\frac{sin(x)}{x}$...and knowing your style you will protest and say you need to just know its 1...but what if you didnt know that? you could do a rate of decreasing...but it is just much more in my opinion beautiful to say $\lim_{x \to 0}\frac{sin(x)}{x}=\lim_{x \to 0}\frac{cos(x)}{1}=1$...there your blatantly sardonic question is answered.
• April 12th 2008, 07:49 AM
Mathstud28
Oh yeah
Quote:

Originally Posted by Moo
Hello,

Looking at which frequency l'Hôpital's rule is used, i'd like to know what is its real utility... D'you have examples where we HAVE to use it ? And can't use anything else ?

As a memento :

If $\lim_{x \to a} f(x)=\lim_{x \to a} g(x)=0 \text{\ or \ } \infty$, then :

$\lim_{x \to a} \frac{f(x)}{g(x)}=\lim_{x \to a} \frac{f'(x)}{g'(x)}$

those are not the only indeterminate forms also you can have $\frac{0}{0},\frac{\infty}{\infty},0^0,1^{\infty},0 ^{\infty},\infty\cdot{0}$ as well as others =)
• April 12th 2008, 07:49 AM
Moo
Do you know why it is 1 ?

$\lim_{x \to 0} \frac{\sin(x)}{x}=\lim_{x \to 0} \frac{\sin(x)-\sin(0)}{x-0}=\cos(0)$ as cos is the derivative for sin.

What do you prefer ? Showing that you can learn by heart a rule ? Or showing that you are able to use the definition of the derivative number ?
• April 12th 2008, 07:51 AM
Mathstud28
Ok
Quote:

Originally Posted by Moo
Do you know why it is 1 ?

$\lim_{x \to 0} \frac{\sin(x)}{x}=\lim_{x \to 0} \frac{\sin(x)-\sin(0)}{x-0}=\cos(0)$ as cos is the derivative for sin.

What do you prefer ? Showing that you can learn by heart a rule ? Or showing that you are able to use the definition of the derivative number ?

But now you are getting semanitical...you just did exactly L'hopitals rule except instead of doing just the derivative you used the difference quotient approach...so in other words you just utilized L'hopitals rule?
• April 12th 2008, 07:52 AM
Moo
Quote:

Originally Posted by Mathstud28
those are not the only indeterminate forms also you can have $\frac{0}{0},\frac{\infty}{\infty},0^0,1^{\infty},0 ^{infty},\infty\cdot{0}$ as well as others =)

Thanks, but without pretention, i think that i know quite well these indeterminate forms.

I'll also add that $1^\infty$ is only indeterminate if it is $f(x)^\infty$ when f(x) tends to 1, not equal.

And that you forgot $\infty - \infty$
• April 12th 2008, 07:53 AM
Moo
Quote:

Originally Posted by Mathstud28
But now you are getting semanitical...you just did exactly L'hopitals rule except instead of doing just the derivative you used the difference quotient approach...so in other words you just utilized L'hopitals rule?

Not at all. I used the definition of the derivative. L'Hôpital's rule is just something telling you to derivate.
• April 12th 2008, 07:56 AM
Mathstud28
Quote:

Originally Posted by Moo
Thanks, but without pretention, i think that i know quite well these indeterminate forms.

I'll also add that $1^\infty$ is only indeterminate if it is $f(x)^\infty$ when f(x) tends to 1, not equal.

And that you forgot $\infty - \infty$

that is why I said as well as others
• April 12th 2008, 07:58 AM
Mathstud28
Ok
Quote:

Originally Posted by Moo
Not at all. I used the definition of the derivative. L'Hôpital's rule is just something telling you to derivate.

two things....Unless I looked at it wrong (which you know I do all the time (Itwasntme)) you just differntiated and go the same result...which is L'hopital's...and second of all how do you make the circumflex?
• April 12th 2008, 08:01 AM
Moo
I'm French, so i have the circumflex on my keyboard ;)

Quote:

Unless I looked at it wrong (which you know I do all the time ) you just differntiated and go the same result...
I didn't differentiate. I know a derivative of sin(x). This is the basic definition of a derivate number...
• April 12th 2008, 08:07 AM
Mathstud28
Ok
Quote:

Originally Posted by Moo
I'm French, so i have the circumflex on my keyboard ;)

I didn't differentiate. I know a derivative of sin(x). This is the basic definition of a derivate number...

The only problem I see with that is you would have to say what you said over 1...because you applied l'hopitals rule except instead of doing f'(x) then evaluating at 0 you jsut went straight to $f'(c)=\lim_{x \to c}\frac{f(x)-f(c)}{x-c}$...but what if you were unsure if applying that method didnt yield another indeterminate form? and about the circumflex...
• April 12th 2008, 08:09 AM
angel.white
On the sin x / x , you can also use the squeeze theorem.
• April 12th 2008, 08:11 AM
Moo
Quote:

but what if you were unsure if applying that method didnt yield another indeterminate form?
If i get an indeterminated form with it, i guess i would have one with l'Hôpital's rule too...

Quote:

because you applied l'hopitals rule except instead of doing f'(x) then evaluating at 0 you jsut went straight
Don't you think that the definition of the derivate number came BEFORE l'Hôpital's rule ?
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