Questions in fun-lim

Select	Question
	The value of $\underset{x\to \infty }{\mathop{\lim }}\,\frac{(x+1)(3x+4)}{{{x}^{2}}(x-8)}$ is equal to
	If $f(x) = \begin{cases} \frac{{\sin [x]}}{{[x]}},{\rm{ when }}[x] \ne 0\\ \,\,\,\,\,\,\,\,\,0,{\rm{ when }}[x] = 0 \end{cases}$ where $[x]$ is greatest integer function, then $\underset{x\to 0}{\mathop{\lim }}\,f(x)=$
	If $\underset{n\to \infty }{\mathop{\lim }}\,\frac{1-{{(10)}^{n}}}{1+{{(10)}^{n+1}}}=\frac{-\alpha }{10}$ , then give the value of $\alpha $ is
	The value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{\log [1+{{x}^{3}}]}{{{\sin }^{3}}x}=$
	$\underset{\theta \to 0}{\mathop{\lim }}\,\frac{4\theta (\tan \theta -2\theta \tan \theta )}{{{(1-\cos 2\theta )}^{2}}}$ is
	The value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{{{27}^{x}}-{{9}^{x}}-{{3}^{x}}+1}{\sqrt{5}-\sqrt{4+\cos x}}$ is
	The value of $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{x}^{n}}}{{{x}^{n}}+1}$ where $x<-1$ is
	The value of $\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+...+\frac{1}{(2n-1)(2n+1)}$ is equal to
	The value of the constant $\alpha $ and $\beta $ such that $\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{x}^{2}}+1}{x+1}-\alpha x-\beta \right)=0$ are respectively
	Let $f:R\to R$ be a differentiable function having $f(2)=6,f'(2)=\left( \frac{1}{48} \right).$ Then $\underset{x\to 2}{\mathop{\lim }}\,\int\limits_{6}^{f(x)}{\frac{4{{t}^{3}}}{x-2}}dt$ equals