For limits-without-lhospital, is the intent to NOT use the definition of the derivative?

Question

The infamous limits-without-lhospital 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 limits-without-lhospital 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?

Answer 1

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 limits-without-lhopital 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.

Answer 2

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.