The infamous tag.

A lot of these questions are of the form $$\lim_{x \to a}\dfrac{f(x)-f(a)}{x-a}$$ and anyone who's seen the first few chapters of a typical calculus book would know the above is the definition of the derivative of $f$ at $a$ when the limit exists, learned much earlier than when L-Hospital's rule is covered.

However, it has been pointed out several times that using this definition of the derivative and solving via L-Hospital's rule are equivalent. Thus, it's ambiguous whether or not using the definition of the derivative is permissible.

For example, after a quick search, I was able to find:

Find the limit of $\lim_{x\to0}{\frac{\ln(1+e^x)-\ln2}{x}}$ without L'Hospital's rule (best answer uses derivative definition)

Compute $\lim_{u\to 0}\frac{(u+1)^\tau-1}{u}$ without l'Hopital ($\tau>0$).

Finding $\lim_{x \to \infty} x(\ln(1+x) - \ln(x))$ without l'Hopital

Compute a limit without L'Hopital's rule $\lim_{x \to a} \frac{a^x-x^a}{x-a}$ (highest-voted answer used derivative definition)

Limit $\lim\limits_{x \to π/2}\frac{2x\sin(x) - π}{\cos x}$ without l'hospital

Finding $\lim_{h \to 0} \frac{(x+h)^4 -x^4}{h}$

My questions are...

  1. Is it worth deciding, as a community, whether or not using the definition of the derivative is permissible for answers to questions?
  2. If the answer to 1 is yes, could we point out in the tag wiki that such questions are not to be answered using the definition of the derivative?
  • 2
    Well, the two methods are not, strictly speaking, equivalent. L'Hôpital's rule requires the function to be continuously differentiable at that point. – Trevor Gunn Dec 26 '17 at 15:29
  • @TrevorGunn Strictly speaking, that's true - however, if you're able to find a limit using the derivative definition, you're assuming that the function is continuously differentiable (I think... my analysis is a bit rusty). – Clarinetist Dec 26 '17 at 15:32
  • 9
    No, if you use the definition of the derivative you are only assuming it is differentiable at that point (by definition). – Trevor Gunn Dec 26 '17 at 15:34
  • 5
    As far as I remember from my own course work, limits without L'Hospital usually meant using two well known limits: $\lim_{x \to 0} \frac {\sin x }{x}$ and the usual limit definition of the exponent. They were introduced before the definition of the derivative. From my experience here most of such questions can be solved by reducing them to one of these limits. – Yuriy S Dec 26 '17 at 16:22
  • Well the fact that you can identify a given limit as a derivative is helpful only when you are allowed to use rules of derivatives and derivatives of elementary functions. For a typical limit problem, the idea is to use limit laws and certain well known limits, rather than going to the derivatives. – Paramanand Singh Dec 27 '17 at 9:23
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    Also the idea behind banning the use of L'Hospital's Rule in such questions is probably to take care of that bad habit of "plugging and if needed defferentiating and plugging" while evaluating limits. – Paramanand Singh Dec 27 '17 at 9:28
  • 2
    "However, it has been pointed out several times that using this definition of the derivative and solving via L-Hospital's rule are equivalent." The comments (which I wrote) of the very first link you give actually state they are not equivalent. – Clement C. Dec 27 '17 at 15:01
  • I just want to note (as someone who suggested the tag in the first place) that there was a problem with people posting answers to questions in which OP stated "without L'Hospital". That very rarely happens now. Also, even the basic limit $\frac {\sin x}x$ has its issues - GH Hardy in Pure Mathematics poses the question "what is the $x$?" and suggests that natural geometric intuition made rigorous requires a notion of arc length or area, or the use of a series (motivation?) - so this is not such an elementary limit as is sometimes assumed. – Mark Bennet Jan 3 at 20:48
up vote 11 down vote accepted

Observe that for these questions (among others), the accepted answer uses the definition of the derivative:

Find the limit of $\lim_{x\to0}{\frac{\ln(1+e^x)-\ln2}{x}}$ without L'Hospital's rule

Finding $\lim_{h \to 0} \frac{(x+h)^4 -x^4}{h}$

For other questions, using the definition of the derivative is not an acceptable answer. One recurring example is for limits that occur in deriving what the derivative of a particular function, e.g.

$$ \lim_{x \to 0} \frac{e^x - 1}{x}.$$

For this particular limit, more context is needed regarding the definition of $e^x$.

It seems clear to me that there are two types of questions that use the tag. I would say, some people get questions assigned as homework (they often don't tell you this) where the solution their instructor has in mind is to use the definition of the derivative. Other people have questions where this is not the intended solution—or perhaps, even, there is no intended solution. The only good way to discern whether or not using the definition of the derivative is acceptable or not is for the person asking the question to provide context.

  • 1
    I agree, mostly. I think the second example is not so good, as the accepted answer indeed provides both types of answers. Moreover, given the notation there I would say it is typically the intent to obtain the derivative from first principles. (The accept is some indication, but there are also "wrong" accepts.) – quid Dec 26 '17 at 15:55
  • "The only good way to discern whether or not using the definition of the derivative is acceptable or not is for the person asking the question to provide context." Perhaps a dumb suggestion... would it be worthwhile providing an alert to provide such context when a question is tagged with limits-without-lhospital? – Clarinetist Dec 26 '17 at 17:21
  • @Clarinetist At that rate, there are probably way to many questions I think need more context. I'd rather ask for clarification on a case-by-case basis. I also think that providing an alert such as that would just get ignored/annoy people unnecessarily. – Simply Beautiful Art Dec 26 '17 at 20:06
  • Perhaps a tag to evaluate derivatives without the definition (the limit of the difference quotient) should be created. – Chase Ryan Taylor Jan 2 at 5:25

On its face, "prove XXX without using YYY" is nonsense. Either the preconditions hold for using theorem YYY, in which case it can be used regardless of the instructions given (the student could always prove theorem YYY using whatever tools are given instead, as a self-contained lemma in the problem answer). Or they do not, and YYY cannot be used, regardless of the instructions.

Of course, there are pedagogical reasons for "without L'Hopital" and similar instructions:

  • The value of the limit is needed in order to prove differentiability of the function in question, as one step in a proof program that the instructor has in mind but wasn't shared by the student;
  • The instructor wants the students to practice reasoning and building intuition about limits rather than blindly plug-and-chug;
  • Using L'Hopital's rule is actually incorrect, perhaps for subtle reasons, and the instructor wants to prevent students from going down a garden path; a favorite example is $$\lim_{x\to \infty} \frac{x+\sin x}{x + \cos x}.$$

However in all of these cases, instead of the dubious "without using L'Hopital" instruction, a superior alternative would be for the question to explain why L'Hopital cannot be used, and this is what I recommend probing in cases of questions of this form. The appropriate way to write the answer will become obvious once this information has been ascertained.

Now if the answer is "because you cannot assume differentiability of this function" and the function is obviously differentiable everywhere (as the composition of differentiable elementary functions), and the other preconditions of L'Hopital's rule hold, then frankly, just use L'Hopital. I'm sorry, but I don't see the value in math.se pedaling suspect mathematics (insisting on there being some important difference between using Taylor's theorem or L'Hopital's rule for solving a run-of-the-mill limit) to accommodate an instructor's poorly-formulated problem set.

  • I think third reason in your answer is most appropriate. I have seen many users here who blindly use this technique and get the right answer without checking if the rule is applicable or not. And most beginners don't even know if there are any conditions to check. For them it's just "differentiate and plug" thing. – Paramanand Singh Jan 4 at 17:13

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