I'd someone to check my work:
Using the definition of the convergence of a sequence, verify that the following converge to the limit that is proposed.
NOTE, the definition of convergence for a sequence is as follows: A sequence (a_n) converges to if, for every pos. number epsilon, there exists an such that whenever n >= N, it follows that |a_n - a| < epsilon.
1.) lim(1/(6n^2 + 1)) = 0
2.) lim((3n+1)/(2n+5) = 3/2
3.) lim(2/sqrt(n+3)) = 0
1.) 1/(6n^2 + 1) < 1/(6n^2) < epsilon => n > sqrt(1/(6*epsilon))
2.) |(3n+1)/(2n+5) - 3/2| = |[2*(3n+1) - 3*(2n+5)]/[2*(2n+5]| = 13/2 * 1/(2n + 5) < 13/4n < epsilon => n > 13/4*epsilon
3.) 2/sqrt(n+3) < 2/sqrt(n) < epsilon => n > (2/epsilon)^2
There we go.. my professor doesn't like that I didn't take it back and show |a_n - a| < epsilon or something like that.. not quite sure what he wants.