# Using the definition of convergence, how do you prove that the sequence  lim (3n+1)/(2n+5)=3/2 converges?

I assume meant limit at infinity, that is, to show that ${\lim}_{n \to \infty} \frac{3 n + 1}{2 n + 5} = \frac{3}{2}$. Note that ${\lim}_{n \to \infty} \frac{1}{n} = 0$ and similarly ${\lim}_{n \to \infty} \frac{k}{n} = 0$ for any real, positive $k$.
Also note that by dividing by $n$ on both numerator and denominator, $\frac{3 n + 1}{2 n + 5} = \frac{3 + \frac{1}{n}}{2 + \frac{5}{n}}$
This means that ${\lim}_{n \to \infty} \frac{3 n + 1}{2 n + 5} = {\lim}_{n \to \infty} \frac{3 + \frac{1}{n}}{2 + \frac{5}{n}} = \frac{3 + 0}{2 + 0} = \frac{3}{2}$