How to improve answer

Question

When I posted an answer from my another profile, it reached, in the first two seconds after posting, very bad score. Of course, it is always possible that someone simply dislike the answer, or the poster, or nickname, or anything. However, reaching score of -3 in the first few seconds after posting is something very concerning me.

This is the answer I'm talking about. I'm trying to see what is wrong in the answer, but I still have no idea. If there is some typo, someone usually leaves a comment without voting. However, very negative score may be reached only if the poster of the answer shows completely misunderstanding of the problem or provides completely useless information.

I would like to improve the answer, but without comment I really have no idea what people see so horrible in the answer. I tried to check if I maybe hhave made some miscalculations, but I think it is correct. Therefore, I'm not sure what to improve.

Why I've asking this here: simply because I don't see what is wrong with the answer and I would really like to improve it. If it turns out that nothing is wrong, but people just simply dislike that answer, I would leave it there, no matter how bad score is. But, I think it would be better to improve if something can be improved.

I would ask you to take a look at the answer and tell me what is wrong with it. Does it provide useless information or does it contain wrong math formula or wrong calculations?

Thank you in advance.

Answer 1

I would ask you to take a look at the answer and tell me what is wrong with it. Does it provide useless information or does it contain wrong math formula or wrong calculations?

Here we go.

Because $x^x$ has it's limit at $x\to0$ (and the limit is $1$), it can be written as

$$\begin{align}\lim_{x\to 0^+}{x^{x^x} -1}&=0^{\lim_{x\to 0^+}{{x^x} }}-1\\&=0^1-1\\&=0-1\\&=-1\end{align}$$

First, there are grammatical mistakes in the first sentence: it should be "its limit", not "it's limit". One could say something like "limit as $x\to 0$" or "continuity at some point", but "limit at $x\to 0$" is rather strange. Moreover, "$x\to 0$" should really be $x\to 0+$. So you could simply say $$ \lim_{x\to 0+}x^x=1. $$

Note that when you prove this limit later, "proof" is a noun, not a verb.

Important: Notice that in this case the limit can be split into two limits just because both $x$ and $x^x$ have finite limits at $x\to0^+$ and because they do not form indeterminate expression (see the edit below, I'm sorry if someone misunderstood what I was trying to say).

(...)

If we have any two functions $f$ and $g$ such that $$\lim_{x\to x_0}f(x)^{g(x)}$$ then it is indeed equal to $$\left(\lim_{x\to x_0}f(x)\right)^{\lim_{x\to x_0}g(x)}$$ if both $f(x)$ and $g(x)$ have finite limit at $x\to x_0$ and if they do not form indeterminate expression. That is what I have said at the beginning of my answer.

I don't think I understand what you are talking about by "If we have any two functions $f$ and $g$ such that $\lim_{x\to x_0}f(x)^{g(x)}$". Did you notice that "$\lim_{x\to x_0}f(x)^{g(x)}$" is not a statement? Moreover, $f(x)^{g(x)}$ is not necessarily defined for arbitrary $f$ and $g$.

One can guess though, you are referring to the following statement:

Note that $a^b:=e^{b\ln a}$ for $a>0$. Suppose $f$ and $g$ are two functions such that both $\lim_{x\to x_0}f(x)$ and $\lim_{x\to x_0}g(x)$ exist. Moreover $f$ is strictly positive and $\lim_{x\to x_0}f(x)>0$. Then $$ \lim_{x\to x_0}f(x)^{g(x)}=(\lim_{x\to x_0}f(x))^{\lim_{x\to x_0}g(x)}. $$

Saying that this is true "if they do not form indeterminate expression" seems begging the question: one could ask "why is that so"? Logical correctness does not substitute clarity. Giving a proof of the statement or at least a sketch of it would be much more convincing than a lengthy "mumble jumble".