Troubles with the Definition of a Limit of a Function

[The Student]

I am quite sure that my answer in the following example is wrong even though it seems to meet the criteria of x > N ⇒|f(x)−L| < ɛ. For the function 1/x (as x → ∞), I seem to be able to use any ɛ > 0. So, when I use N = ɛ = 5, it seems to work for x > N ⇒|f(x)−L| < ɛ. But my notes have N = 1/ɛ as a sufficient solution for N. I know my solution doesn't produce a minimal ɛ, but does it work? Why or why not?

 

[pka]

I am quite sure that my answer in the following example is wrong even though it seems to meet the criteria of x > N ⇒|f(x)−L| < ɛ. For the function 1/x (as x → ∞), I seem to be able to use any ɛ > 0. So, when I use N = ɛ = 5, it seems to work for x > N ⇒|f(x)−L| < ɛ. But my notes have N = 1/ɛ as a sufficient solution for N. I know my solution doesn't produce a minimal ɛ, but does it work? Why or why not?

Look I have already responded to this once before.
​The positive integers are not bounded above.
If \(\displaystyle \varepsilon > 0\) then \(\displaystyle \frac{1}{\varepsilon }\) is not an upper bound for \(\displaystyle \mathbb{Z}^+\).
So \(\displaystyle \exists n\in\mathbb{Z}^+[~n\ge\frac{1}{\varepsilon }~]\) Thus \(\displaystyle \varepsilon \ge \frac{1}{n}\)

 

[The Student]

Look I have already responded to this once before.
​The positive integers are not bounded above.
If \(\displaystyle \varepsilon > 0\) then \(\displaystyle \frac{1}{\varepsilon }\) is not an upper bound for \(\displaystyle \mathbb{Z}^+\).
So \(\displaystyle \exists n\in\mathbb{Z}^+[~n\ge\frac{1}{\varepsilon }~]\) Thus \(\displaystyle \varepsilon \ge \frac{1}{n}\)

Is there anything in the definition x > N ⇒|f(x)−L| < ɛ that implies my answer does not work? Here I will just fill in my answer to the definition x < 5 ⇒ 1/x < 5. N = ɛ = 5 seems to work in the definition, doesn't it?

 

[pka]

Is there anything in the definition x > N ⇒|f(x)−L| < ɛ that implies my answer does not work? Here I will just fill in my answer to the definition x < 5 ⇒ 1/x < 5. N = ɛ = 5 seems to work in the definition, doesn't it?

I have absolutely no idea what you are asking there.

State the problem exactly. Show your effort to solve it.