Question

So I have this ques in limits, for example in (n^2+n)/ n^2 , and n approaches infinity. How is the answer : 1. Please need help.

Answer 1

as n approaches infinity, n becomes insignificant compared with n^2, so we get:\[\lim_{n\rightarrow\infty}\frac{n^2+n}{n^2}=\frac{n^2}{n^2}=1\]another way to look at this is to first simplify the expression as follows:\[\frac{n^2+n}{n^2}=\frac{n(n+1)}{n^2}=\frac{n+1}{n}\]now as n approaches infinity, adding one to it becomes insignificant so that n+1 tends to n as n approaches infinity. thus we get:\[\lim_{n\rightarrow\infty}\frac{n+1}{n}=\frac{n}{n}=1\]