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Math Help - L'Hospital's rule

  1. #1
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    L'Hospital's rule

    Ref: Challenging limit

    L'Hospital's rule derives from the Mean Value Theorem.

    Reference thread won't let me respond.
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    Re: L'Hospital's rule

    Quote Originally Posted by Hartlw View Post
    Ref: Challenging limit

    L'Hospital's rule derives from the Mean Value Theorem.

    Reference thread won't let me respond.
    I'm not sure I see your point. If the Mean Value Theorem was derived from L'Hospital's Rule, then I could see a problem (since L'Hospital's Rule would be needed to prove the Mean Value Theorem). Since it is the other way around, and the Mean Value Theorem is used in the proof of L'Hospital's Rule, then just because I am using the Mean Value Theorem does not mean I am applying L'Hospital's Rule.
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    Re: L'Hospital's rule

    It's a matter of the spirit of topsquark's rules, whether using the method to derive L'Hospitals rule is not using L'Hospital's rule. But I see your point.

    Just to be sure it wasn't some kind of fluke, i checked for x=.0001 on a calculator and sure enough, .16666 = 1/6.
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    Re: L'Hospital's rule

    You can derive L'Hospital's Rule using other methods besides the Mean Value Theorem. I myself like using Taylor Series.
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    Re: L'Hospital's rule

    Quote Originally Posted by Prove It View Post
    You can derive L'Hospital's Rule using other methods besides the Mean Value Theorem. I myself like using Taylor Series.
    Which derive from the mean value theorem.

    EDIT: But you're not supposed to use series in the referenced limit.
    In any event, it would be topsquark's call as to what is admissible. I'm not judging, just observing.

    My only interest in the question is an answer that is:
    1) simple and easy to follow, ie, that I can understand. My personal preference, not a judgement.
    2) educational.
    Last edited by Hartlw; June 12th 2014 at 05:58 AM.
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    Re: L'Hospital's rule

    Quote Originally Posted by Hartlw View Post
    Which derive from the mean value theorem.
    I take your point. I suppose a logical question would be, does L'Hospital's Rule imply the Mean Value Theorem? In other words, if we assume L'Hospital's Rule is true, can we prove the Mean Value Theorem? Since the Mean Value Theorem involves the existence of a point at which some equation holds, there are no limits to evaluate using L'Hospital's Rule. So, I don't believe L'Hospital's Rule implies the Mean Value Theorem. If there is a way to show L'Hopsital's Rule implies the Mean Value Theorem, then I agree that we should avoid using the Mean Value Theorem, as well.

    On the other hand, if L'Hospital's Rule does not imply the Mean Value Theorem, then we should only avoid its use if we are also avoiding everything upon which L'Hospital's Rule is derived. Suppose we list everything used to prove L'Hospital's Rule (in order to avoid using the same in the evaluation of the original limit). L'Hospital's rule is derived from a sequence of logical statements (any mathematical proof is a sequence of logical statements). So, our list contains the system of logic that is the basis for all mathematical proof. Without logic, the original limit is rather moot.
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    Re: L'Hospital's rule

    Quote Originally Posted by Hartlw View Post
    Which derive from the mean value theorem.

    EDIT: But you're not supposed to use series in the referenced limit.
    In any event, it would be topsquark's call as to what is admissible. I'm not judging, just observing.

    My only interest in the question is an answer that is:
    1) simple and easy to follow, ie, that I can understand. My personal preference, not a judgement.
    2) educational.
    I've never used the Mean Value Theorem to derive Taylor Series. The only time I can think of myself using it with anything to do with Taylor Series is getting an expression for the remainder, which you don't need to do if you don't truncate...
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    Re: L'Hospital's rule

    Quote Originally Posted by Hartlw View Post
    Which derive from the mean value theorem.

    EDIT: But you're not supposed to use series in the referenced limit.
    In any event, it would be topsquark's call as to what is admissible. I'm not judging, just observing.

    My only interest in the question is an answer that is:
    1) simple and easy to follow, ie, that I can understand. My personal preference, not a judgement.
    2) educational.
    Because the mean value theorem came up in response to my question, I can say that I did find its use educational. Admittedly, what is educational is always relative to one's ignorance. Idea's answer was certainly simple, but I personally could not follow all of it. The explanation using the Mean Value Theorem meant I could follow Idea's solution.
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    Re: L'Hospital's rule

    Oh well, what the heck.

    To the other posters:

    What is your definition of sinx?
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    Re: L'Hospital's rule

    Hmmmm. How do you do the problem without a definition of sinx?

    But anyhow, hereís whatís bothering me.

    Without series, the definition of sinx is a geometric one. So I thought, ah ha, there is a geometric solution to the problem: draw unit circle used to define sinx (x is angle in radians, sin x is y coordinate of ray), and just write down some obvious geometric relations. No luck, but you can show a limit exists.

    OK start small. Think about lim sinx/x= 1. Itís a fairly simple standard calculus geometric proof. Can you use LíHospitalís rule for this? No, because the definition of derivative of sinx depends on lim sinx/x. But no need to worry, because the geometric proof works fine for lim sinx/x. But I couldnít get a geometric proof to work for current problem.

    On the way I note that LíHospitalís rule doesnít work for (1-sinx/x)/x2 because you canít get derivative of sinx/x (try it). But no problem using LíHospitals rule with (x-sinx)/x3.
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    Re: L'Hospital's rule

    Forget previous comment about derivative of sinx/x not existing. I meant to say you canít find lim (1-sinx/x)x2 by Líhospitalís rule.
    I finally decided (Iím trying to get over prejudice from past experience) to read Slipeternals post #7 of OP reference, which turned out to be quite straightforward. All it did was show f(x) is monotonically increasing on (0,2pi). Combining that with the previous post #4 of OP ref gives,


    1/6 ≤ lim (1-sinx/x)/x2 < 1/6,

    Which is clearly wrong. So the challenge problem has not yet been solved. Personally I think the real challenge is to start from scratch and derive the 1/6, which I still believe should be possible geometrically, like lim sinx/x.

    It really is a pain to be locked out of a thread, the OP reference in this case. As a courtesy, the management should announce I am locked out so as not to give the impression I accept the results of the thread, which I clearly donít.
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    Re: L'Hospital's rule

    I accept SlipEternals conclusion:

    1) lim (1-sinx/x)/x2 < 1/6

    Yes, it was nice, and completeley in line with the requirements of the challenge, invalidating my questioning in the above posts.

    I assumed the conclusion was based on the sandwich theorem, which was implied as the path in previous posts of the ref in OP, and I couldn't see that despite a popular misconception which I correct in the following:

    sinx/x and the Squeeze Theorem

    Basically, 1) is correct by the definition of a limit, not by the sandwhich theorem.

    (I still think the appropriate proof is geometric, in keeping with the definition of sinx/x and sine)
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    Re: L'Hospital's rule

    Quote Originally Posted by Hartlw View Post
    I accept SlipEternals conclusion:

    1) lim (1-sinx/x)/x2 < 1/6

    Yes, it was nice, and completeley in line with the requirements of the challenge, invalidating my questioning in the above posts.

    I assumed the conclusion was based on the sandwich theorem, which was implied as the path in previous posts of the ref in OP, and I couldn't see that despite a popular misconception which I correct in the following:

    sinx/x and the Squeeze Theorem

    Basically, 1) is correct by the definition of a limit, not by the sandwhich theorem.

    (I still think the appropriate proof is geometric, in keeping with the definition of sinx/x and sine)
    Oh well, what the heck? Everything "was clearly wrong."
    Last edited by JeffM; June 14th 2014 at 04:19 PM.
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    Re: L'Hospital's rule

    The answer to OP reference is the limit (x-sinx)/x3 = 6 canít be proved without series or LíHospitalís rule because:

    1) You canít find limit analytically because you canít find sinx without series.

    2) You canít find the limit geometrically, as you do with sinx/x, because you canít picture x2 or x3. Geometrically (pictorially), you can add angles but you canít multiply them.

    In retrospect, it is clear that the 6 and the problem come from the series for sinx, not from geometry.

    Would appreciate someone posting this, and referencing it, in the original ďchallenge problemĒ thread:
    Challenging limit
    which I am locked out of.
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