Marco

Wolfram does not find any closed form. This is not a proof that it doesn't exist, but I think it is very unlikely that it does exist

If $A,B$ symmetric and definite then $$ AA^T=BB^T $$ implies $A=\pm B$ As it is shown here. Similarly for any $A,B$ skew symmetric and definite the two matrices are equal up to a sign. So if you meant that the equality holds for all the matrices $X$ then in the symmetric definite case you have equality up to a sign

@TymaGaidash I was thinking about Lagrange inversion formula, but $\operatorname{sinhc}$ isn't actually analytic near the origin, so i don't think it works

Does it have to be done by using the Ramanujan master theorem? Or you are only interested in a closed form for the $\operatorname{sinhc}^{-1}$

Assume $a=2.3$ and $\delta=1$. If $x=3.1$ , $|x-a|<\delta$, but $|f(x)-2|=1>\epsilon$ for each $\epsilon<1$, so $\delta=1$ does not work

@Dam there is an answer to this question and three dupes questions linked to it. The answer posted here is correct. As your notes. You can't use arbitrary $\delta$. e.g. $\delta=10$ does not work

@DanAsimov To be fair, as the floor function is locally constant you can choose $\delta$ fully independent by $\epsilon$

It is not correct to choose any valiue of $\delta$, as the limit does not exists. For the not integer part. The function is constant outside the integer points. Do you know how to find $\lim_{x \to a} g(x)$ when $g(x)=n$ is the constant function?

@DanAsimov it means to calculate the limit in $(-\infty, a)$ (left limit) and $(a, \infty)$ (right limit)

If left and right limit does not coincide the limit does not exists. Your calculation is correct, but the conclusion is not

"I do the left and right hand limits where in both cases $|f(x)−l|=0$" are you sure about that? What does it happen to the left limit when $a$ is an integer?