$\mathop {\lim }\limits_{x \to \infty } \frac{{3{x^2} + 2x - 1}}{{2{x^2} - 3x - 3}} = $

Let $f(x) = \lim_{y \rightarrow \infty} y(x^{1/y} - 1)$,and $2022 f(\frac{1}{x}) + P f(x) = f(x^2)$,then $P =$

$\lim _{x \rightarrow 0} \frac{e^x-e^{\sin x}}{2(x-\sin x)}$

If $f(x) = \begin{cases} x^2-1, & 0 < x < 2 \\ 2x+3, & 2 \leq x < 3 \end{cases}$,the quadratic equation whose roots are $\lim_{x \rightarrow 2^{-}} f(x)$ and $\lim_{x \rightarrow 2^{+}} f(x)$ is

If $\mathop {\lim }\limits_{n \to \infty } \frac{1 - 10^n}{1 + 10^{n+1}} = \frac{-\alpha}{10}$,then find the value of $\alpha$.

$\lim _{x \rightarrow \infty}\left(\frac{x+8}{x+1}\right)^{x+5} = \dots$