Limits

Solve :
$$ \underset { x \rightarrow \infty}{lim} \dfrac{3x^{2}+5x+2}{5x^{2}+6x+1}$$
$$A_i \displaystyle = \frac{x - a_i}{|x - a_i|}, i = 1, 2, ..... , n,$$ and $$a_1 < a_2 < a_3 < ..... < a_n.$$
We know that $$\sin x< x$$ for any $$x> 0$$ & $$\sin x> x$$ for any $$x< 0$$.
Thus for any $$n\epsilon N$$, $$\sin n< n$$. But as $$-1< \sin n< 1$$ &
$$\displaystyle \left ( -1, 1 \right )\subset \left [ -\frac{\pi }{2}, \frac{\pi }{2} \right ]$$, let us define two recursion sequences $$\left \{ a_{n} \right \}$$ & $$\left \{ b_{n} \right \}$$ give as
$$a_{1}=1$$, $$a_{n}=\sin a_{n-1}$$
$$b_{1}=1$$, $$b_{n}=\cos b_{n-1}$$
On the basis of above information answer the following questions.

A sequence is a function whose domain is the set of natural number. A sequence $$s_{1}, s_{2}...$$ of real numbers is said to have a limit $$l$$ if $$\lim_{n\rightarrow \infty }s_{n}=l$$. If $$l< \infty ,$$ then $$< s_n> $$ is said to convergent. The following are well known
(i) $$\displaystyle \lim_{n\rightarrow \infty }\dfrac{1}{n^{p}}=0(p> 0)$$

(ii) $$\displaystyle \lim_{n\rightarrow \infty }x^{n}=0(\left | x \right |< 1)$$

(iii) If $$\displaystyle  \lim_{n\rightarrow \infty }s_{n}=l$$, then $$\displaystyle  \lim_{n\rightarrow \infty}\frac{s_{1}+s_{2}+...+s_{n}}{n}=l$$
Let f(n) denote the  $$n^{th}$$ term of the sequence 3, 6, 11, 18, 27............. and g(n) denote the $$n^{th}$$ term of the sequence 3, 7, 13, 21, .............. let F(n) and G(n) denote the sum of n-terms of the above sequences respectively