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. 

Hints: for an arbitrary $\,\epsilon>0\;$ :
$$\left|\frac{3n^2+4}{2n^2+5}-\frac32\right|=\left|\frac{-7}{2(2n^2+5)}\right|<\epsilon\iff2n^2+5>\frac7{2\epsilon}\iff$$
$$(**)\;\;2n^2>\frac7{2\epsilon}-5\ldots$$
Be sure you can prove you can choose some $\,M\in\Bbb N\;$ s.t. for all $\,n>M\;$ the inequality (**) is true.
