What is the method for finding the limit of a sequence?

[The Student]

My notes ask if there will always be a real number N such that, n > N ⇒ |1/(n^2+n)| < ɛ, n ∈ ℕ. The notes then show 1/(n^2+n) < 1/n^2, which makes sense. Then the notes show the answer N = 1/(ɛ)^1/2. So I am not sure what steps should be taken that will work for other questions like this. My thought process is that we can make ɛ = 1/n^2 so that we have an epsilon that will always be smaller than |1/(n^2+n)|. But how exactly does the N come into this?

 

[pka]

My notes ask if there will always be a real number N such that, n > N ⇒ |1/(n^2+n)| < ɛ, n ∈ ℕ. The notes then show 1/(n^2+n) < 1/n^2, which makes sense. Then the notes show the answer N = 1/(ɛ)^1/2. So I am not sure what steps should be taken that will work for other questions like this. My thought process is that we can make ɛ = 1/n^2 so that we have an epsilon that will always be smaller than |1/(n^2+n)|. But how exactly does the N come into this?

\(\displaystyle N = \dfrac{1}{{\sqrt \varepsilon }} \Rightarrow \;\varepsilon = \dfrac{1}{{{N^2}}} > \dfrac{1}{{{N^2} + N}}\)

If \(\displaystyle n\ge N\) then \(\displaystyle \dfrac{1}{N^2+N}\ge\dfrac{1}{n^2+n}\)

 

[The Student]

\(\displaystyle n = \dfrac{1}{{\sqrt \varepsilon }} \rightarrow \;\varepsilon = \dfrac{1}{{{n^2}}} > \dfrac{1}{{{n^2} + n}}\)

if \(\displaystyle n\ge n\) then \(\displaystyle \dfrac{1}{n^2+n}\ge\dfrac{1}{n^2+n}\)

Thank-you!!!