Example Find the limit of the following sequence.

$\displaystyle \displaystyle{\{a_{n}\} = \{\frac{3n^2 - 5n}{5n^2 + 2n - 6}\}} $

Answer To find the limit of a sequence, it is important to know the followings:

1. The limit of a sum is the sum of the limits. $\lim_{n \to \infty}(a_{n} + b_{n}) = \lim_{n \to \infty}a_{n} + \lim_{n \to \infty}b_{n}$  
2. The limit of a constant multiple is the constant times the limit. $\lim_{n \to \infty}(\alpha a_{n}) = \alpha \lim_{n \to \infty} a_{n}$  
3. $\lim_{n \to \infty}a_{n}b_{n} = \lim_{n \to \infty}a_{n} \lim_{n \to \infty}b_{n}$  
4. $\lim_{n \to \infty}\frac{a_{n}}{b_{n}} = \frac{\lim_{n \to \infty}a_{n}}{\lim_{n \to \infty}b_{n}}$. ただし, $\lim_{n \to \infty}b_{n} \neq 0$  
5. Basic formula for the limit. $\lim_{n \to \infty}(1 + \frac{1}{n})^{n} = e$  

Apply these rules to the above problem, we have First, as $n \rightarrow \infty$, we have $\displaystyle{3n^2 - 5n = n^2 (3 - \frac{5}{n})}$ $\longrightarrow \infty$,Also, $5n^2 + 2n - 6 =$ $\displaystyle{n^2 (5 + \frac{2}{n} - \frac{6}{n^2}) \longrightarrow \infty}$.Then we factor out the largest degree term which is $n^2$. Then

$\displaystyle \lim_{n \rightarrow \infty}\frac{3n^2 - 5n}{5n^2 + 2n - 6} = \lim_{n \rightarrow \infty}\frac{3 - 5/n}{5 + 3/n - 6/n^2} = \frac{3}{5}. $