A Question about the Definition of the Limit of a Function

[The Student]

I know how to use 0 < |x−a| < δ ⇒|f(x)−L| < ɛ when "a" is some fixed real number, but what do we do when "a" = ∞? Can we still use the definition 0 < |x−a| < δ ⇒|f(x)−L| < ɛ?

 

[pka]

I know how to use 0 < |x−a| < δ ⇒|f(x)−L| < ɛ when "a" is some fixed real number, but what do we do when "a" = ∞? Can we still use the definition 0 < |x−a| < δ ⇒|f(x)−L| < ɛ?

We use the fact that the positive integers are not bounded above.
The statement that \(\displaystyle {\lim _{x \to \infty }}f(x) = L\) means that
\(\displaystyle \left( {\forall \varepsilon > 0} \right)\left( {\exists N \in {Z^ + }} \right)\left[ {x \ge N \Rightarrow \;\left| {f(x) - L} \right| < \varepsilon } \right]\)

 

[The Student]

We use the fact that the positive integers are not bounded above.
The statement that \(\displaystyle {\lim _{x \to \infty }}f(x) = L\) means that
\(\displaystyle \left( {\forall \varepsilon > 0} \right)\left( {\exists N \in {Z^ + }} \right)\left[ {x \ge N \Rightarrow \;\left| {f(x) - L} \right| < \varepsilon } \right]\)

Thank-you!