$\mathop {\lim }\limits_{n \to \infty } {({4^n} + {5^n})^{1/n}}$ is equal to

The value of the limit $\lim _{x \rightarrow-\infty}\left(\sqrt{4 x^2-x}+2 x\right)$ is

If a function $f$ is defined by $f(x) = \frac{\cot^3 x - \tan x}{\cos(x + \pi/4)}$ for $x \neq \pi/4$,then $\lim_{x \rightarrow \pi/4} f(x) = $

$\lim _{n}$ ${\rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\ldots +\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\ldots +n^3\right)-\left(1^2+2^2+\ldots +n^2\right)}$ is equal to:

$\lim_{x \rightarrow -\infty} \frac{3|x|-x}{|x|-2x} - \lim_{x \rightarrow 0} \frac{\log(1+x^3)}{\sin^3 x} =$

If $a$ is the minimum value of $\sin^2 \theta - \sin \theta + \frac{1}{2}$ and $b = \lim_{x \to \infty} (\sqrt{(x + 1)(x + 2)} - x)$,then $|2a + b| = $