Finding out how out how much is 100% I know 14%

Fractions with radicals in the denominator
I'm working my way through the videos on the Khan Academy, and have a hit a road block. I can't understand why the following is true:
${\displaystyle \frac{6}{\frac{6\sqrt{85}}{85}}=\sqrt{85}}$

What is the value of? ${\displaystyle \frac{1}{3}÷4}$

What is ${\displaystyle \frac{11}{14}}$ divide by ${\displaystyle \frac{5}{6}}$ in simple form

Prove ${\displaystyle \frac{1}{4n^2-1}=\frac{1}{(2n+1)(2n-1)}=\frac{1}{2(2n-1)}-\frac{1}{2(2n+1)}}$
Could you explain the operation in the third step?
${\displaystyle \frac{1}{4n^2-1}=\frac{1}{(2n+1)(2n-1)}=\frac{1}{2(2n-1)}-\frac{1}{2(2n+1)}}$
It comes from the sumation
${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{4n^2-1}}$
I could just copy it to my homework, but I'd like to know how this conversion works. Thanks in advance.

How do you simplify ${\displaystyle \frac{(n+2)!}{n!}?}$

How to show that show that $\frac{v+u}{1+uv/c^2}=c$ when $u=c$?
Context
It's needed for a physics proof that I'm working on. This is the formula for relative velocity, u represents the velocity of the observer, v represents the velocity of a projectile and c represents the speed of light.
Einstein said that this is right, so I'm sure that it is I'm just not sure how it simplifies.

limit of floor function, ceiling function and fractions
We know that
$\lfloor x\rfloor$ : greatest integer $\le x$
$\lceil x\rceil$ : least integer $\ge x$
$\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}(x)=x-\lfloor x\rfloor$
I can solve the questions limit of function like
$$\begin{gathered} \underset{x\to n^{\pm }}{lim}\frac{\lfloor x-1\rfloor }{x-1} \\ \underset{x\to n^{\pm }}{lim}\frac{\lfloor x\rfloor }{x-1} \end{gathered}$$
As $x$ approaches $n$ from above, $\lfloor x-1\rfloor =n-1$; therefore,
$\underset{x\to n^{+}}{lim}\frac{\lfloor x-1\rfloor }{x-1}=\frac{n-1}{n-1}$
but I can't solve the questions like
$$\begin{gathered} \underset{x\to \mathrm{\infty }}{lim}\frac{\lfloor x-3\rfloor }{x-1}, \\ \underset{x\to \mathrm{\infty }}{lim}\frac{\lceil x-3\rceil }{x-1}, \\ \underset{x\to a}{lim}\lfloor x\rfloor , \\ \underset{x\to a}{lim}\lceil x\rceil , \\ \underset{x\to a}{lim}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}(x) \end{gathered}$$
for $a\in \mathbf{R}$
All questions are similar type, so I have given many problems in my questions. Please help me.