### Limits

goals
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