I am currently learning series and testing for convergence. For comparison tests especially I am having an issue grasping the concept of picking a proper limit to compare too. For example the following problem If someone could please put it in the form where it actually looks like what it does on paper that would be great. I am not quite sure how to do that yet. (sqrt(n^2-1)/(n^3+2n^2+5) In the solutions my teacher provided it states to use the comparison test with 1/n^2 and since that is convergent than the original being less all the time is also convergent. I understand that part, I just don't get how to figure out how 1/n^2 was chosen. Another one is limit comparison for sin(1/n)/n. This one I am clueless about and do not understand the solution at all. The solutions state to compare to 1/n^2 and then it states lim n→∞ (sin(1/n)/(n(1/n^2)) and then that becomes lim n→∞ an/bn = 1/n^2/n^2 which is n^2/n^2 = 1 and that would mean its convergent. I know how to tests work but just not how these comparison limits are chosen and especially for the second one the solution is not clear how to get from one step to the other. Thanks in advance.