Prove that the sequence $(3n^2+4)/(2n^2+5) $ converges to $3/2$ 
Prove directly from the definition that the sequence $\left( \dfrac{3n^2+4}{2n^2+5} \right)$ converges to $\dfrac{3}{2}$.

I know that the definition of a limit of a sequence is $|a_n - L| < \varepsilon$ .
However I do not know how to prove this using this definition. 
Any help is kindly appreciated. 

Suppose that it does not converge to $3/2$. Then you have that there exists $\varepsilon_0>0$ such that for every $n\in\mathbb N$ we have $|\displaystyle \frac {3n^2+4}{2n^2+5} - \frac {3}{2}|\geq \varepsilon_0 $ which is equivalent to $\displaystyle \frac {7}{4n^2+10}\geq \varepsilon_0$ for every $n\in\mathbb N$ which is not true since we can make the fraction as small as we like so the sequence is convergent and the limit is $3/2$.
