$\lim _{x \rightarrow \infty} (\sqrt{x^2+5x-7}-x) = $

$\mathop {\lim }\limits_{x \to 0} \frac{x}{|x| + {x^2}} = $

The value of $\mathop {\lim }\limits_{x \to \infty } \frac{(x + 1)(3x + 4)}{x^2(x - 8)}$ is equal to

$\lim _{x \rightarrow 0} \frac{\sin |x|}{x}$ is equal to

Let $f(x)=5-|x-2|$ and $g(x)=|x+1|, x \in R$. If $f(x)$ attains its maximum value at $\alpha$ and $g(x)$ attains its minimum value at $\beta$,then $\lim _{x \rightarrow-\alpha \beta} \frac{(x-1)\left(x^2-5 x+6\right)}{x^2-6 x+8}$ is equal to

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,