The value of $\lim _{x \rightarrow 0} \frac{15^{x}-5^{x}-3^{x}+1}{1-\cos 2 x}$ is

Consider the following statements:
$I$. $\lim _{n \rightarrow \infty} \frac{2^n+(-2)^n}{2^n}$ does not exist.
$II$. $\lim _{n \rightarrow \infty} \frac{3^n+(-3)^n}{4^n}$ does not exist.
Then,

The value of the limit $\lim_{x \rightarrow 1} \frac{\sin(e^{x-1}-1)}{\log x}$ is

The set of all values of $a$ for which $\lim_{x \rightarrow a}(\lfloor x-5 \rfloor - \lfloor 2x+2 \rfloor) = 0$,where $\lfloor \alpha \rfloor$ denotes the greatest integer less than or equal to $\alpha$,is equal to

For each $x \in \mathbb{R}$,let $[x]$ be the greatest integer less than or equal to $x$. Then $\lim_{x \to 0^+} \frac{x([x] + |x|) \sin [x]}{|x|}$ is equal to

If $A \neq 0$ and $x > 0$,then $\lim _{n \rightarrow \infty} \frac{\cos x - e^{nx}}{1 - A e^{nx}} = $