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Math Help - About the real utility of l'H˘pital's rule

  1. #1
    Moo
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    About the real utility of l'H˘pital's rule

    Hello,

    Looking at which frequency l'H˘pital'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)}
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    Super Member angel.white's Avatar
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    Quote Originally Posted by Moo View Post
    Hello,

    Looking at which frequency l'H˘pital'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.
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    Moo
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    Quote Originally Posted by angel.white View Post
    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.
    I looooove your answer !

    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.
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    MHF Contributor Mathstud28's Avatar
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    Ok

    Quote Originally Posted by Moo View Post
    Hello,

    Looking at which frequency l'H˘pital'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.
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    MHF Contributor Mathstud28's Avatar
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    Oh yeah

    Quote Originally Posted by Moo View Post
    Hello,

    Looking at which frequency l'H˘pital'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 =)
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    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 ?
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    MHF Contributor Mathstud28's Avatar
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    Ok

    Quote Originally Posted by Moo View Post
    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?
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    Moo
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    Quote Originally Posted by Mathstud28 View Post
    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
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    Moo
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    Quote Originally Posted by Mathstud28 View Post
    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'H˘pital's rule is just something telling you to derivate.
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Moo View Post
    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
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    MHF Contributor Mathstud28's Avatar
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    Ok

    Quote Originally Posted by Moo View Post
    Not at all. I used the definition of the derivative. L'H˘pital'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 ) you just differntiated and go the same result...which is L'hopital's...and second of all how do you make the circumflex?
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    I'm French, so i have the circumflex on my keyboard

    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...
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    MHF Contributor Mathstud28's Avatar
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    Ok

    Quote Originally Posted by Moo View Post
    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...
    Last edited by ThePerfectHacker; April 12th 2008 at 06:59 PM.
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    Super Member angel.white's Avatar
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    On the sin x / x , you can also use the squeeze theorem.
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    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'H˘pital's rule too...

    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'H˘pital's rule ?
    Last edited by ThePerfectHacker; April 12th 2008 at 06:59 PM.
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