Questions in fun-lim

Select	Question
	$\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{{{n}^{3}}+1}+\frac{4}{{{n}^{3}}+1}+\frac{9}{{{n}^{3}}+1}+........+\frac{{{n}^{2}}}{{{n}^{3}}+1} \right]=$
	If ${{S}_{n}}=\sum\limits_{k=1}^{n}{{{a}_{k}}}$ and$\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=a,$ then $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{S}_{n+1}}-{{S}_{n}}}{\sqrt{\sum\limits_{k=1}^{n}{k}}}$ is equal to
	If ${{a}_{1}}=1$ and ${{a}_{n+1}}=\frac{4+3{{a}_{n}}}{3+2{{a}_{n}}},\ n\ge 1$ and if $-\frac{1}{3}$ , then the value of a is
	The value of $\underset{n\to \infty }{\mathop{\lim }}\,\cos \left( \frac{x}{2} \right)\cos \left( \frac{x}{4} \right)\cos \left( \frac{x}{8} \right)...\cos \left( \frac{x}{{{2}^{n}}} \right)$ is
	$\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+...+\frac{1}{{{2}^{n}}}$ equals
	$\underset{n\to \infty }{\mathop{\lim }}\,\left\{ \frac{1}{{{n}^{2}}}+\frac{2}{{{n}^{2}}}+\frac{3}{{{n}^{2}}}+......+\frac{n}{{{n}^{2}}} \right\}$ is
	The value of $\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{1-{{n}^{2}}}{\sum n}$ will be
	If ${{x}_{n}}=\frac{1-2+3-4+5-6+.....-2n}{\sqrt{{{n}^{2}}+1}+\sqrt{4{{n}^{2}}-1}},$ then $\underset{n\to \infty }{\mathop{\lim }}\,{{x}_{n}}$ is equal to
	$\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{(x+1)}^{10}}+{{(x+2)}^{10}}+.....+{{(x+100)}^{10}}}{{{x}^{10}}+{{10}^{10}}}$ is equal to
	The value of $\underset{n\to \infty }{\mathop{\lim }}\,\frac{1+2+3+....n}{{{n}^{2}}+100}$ is equal