How do I find the limit of a sequence that is expressed as a fraction?

There are a number of ways of looking at the limiting behaviour of a fraction. Let’s look at three examples:1) a(n) = 2n+1/7n —> divide into two separate terms, that both clearly converge. 2) b(n) = 2/( n^2-1) = (2) x (1)/(n+1)(n-1) = (2) ((A/n+1)+(B/n-1)) = (2) ((-1/n+1)+(1/n-1)) —> Partial fractions method with difference of two squares. 3) c(n) = 8n+7 / (x+2)(x-1) = 3/x+2 + 5/x-1 —> Partial fractions (include other rules too).