$\lim _{x}$ ${\rightarrow -a} \frac{x^7+a^7}{x+a} = 7$ $\Rightarrow a = ?$

If $\mathop {\lim }\limits_{x \to 1} {\sin ^{ - 1}}\left( {\frac{k}{{\ln x}} - \frac{k}{{x - 1}}} \right)$ exists, then the number of integers in the range of $k$ is:

If $\lim _{x \rightarrow 0}\left\{1+x \log \left(1+a^2\right)\right\}^{1 / x}=2 a \sin ^2 \theta$,where $a>0$ and $\theta \in R$,then:

Let $a$ be an integer such that $\lim \limits_{x \rightarrow 7} \frac{18-[1-x]}{[x]-3a}$ exists,where $[t]$ denotes the greatest integer function $\leq t$. Then $a$ is equal to:

If $\mathop {\lim }\limits_{x \to \infty } \left\{ {\ln \left( {{x^2} + 5x} \right) - 2\ln \left( {cx + 1} \right)} \right\} = -2$,then:

For $\alpha, \beta, \gamma \in R$,if $\lim _{x \rightarrow 0} \frac{x^2 \sin(\alpha x) + (\gamma-1) e^{x^2}}{\sin(2x) - \beta x} = 3$,then $\beta + \gamma - \alpha$ is equal to: