Question

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

Answer 1

I assume meant limit at infinity, that is, to show that `lim_(n->infty) (3n+1)/(2n+5) = 3/2`. Note that `lim_(n->infty)1/n = 0` and similarly `lim_(n->infty)k/n = 0` for any real, positive `k`.

Explanation:

Also note that by dividing by `n` on both numerator and denominator, `(3n+1)/(2n+5) = (3+1/n)/(2+5/n)`

This means that `lim_(n->infty) (3n+1)/(2n+5) = lim_(n->infty) (3+1/n)/(2+5/n) = (3+0)/(2+0) = 3/2`