Ref: Challenging limit
L'Hospital's rule derives from the Mean Value Theorem.
Reference thread won't let me respond.
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.
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.
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.
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.
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.
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)
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:
which I am locked out of.