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Section1.8(optional) — Making infinite limits a little more formal

For those of you who made it through the formal \(\epsilon-\delta\) definition of limits we give the formal definition of limits at infinity:

Definition1.8.1Limits at infinity — formal

Let \(f\) be a function defined on the whole real line. We say that

the limit as \(x\) approaches \(\infty\) of \(f(x)\) is \(L\)

or equivalently

\(f(x)\) converges to \(L\) as \(x\) goes to \(\infty\)

and write

\begin{align*} \lim_{x \to \infty} f(x) &= L \end{align*}

if and only if for every \(\epsilon \gt 0\) there exists \(M \in \mathbb{R}\) so that \(|f(x)-L| \lt \epsilon\) whenever \(x \gt M\text{.}\)

Similarly we write

\begin{align*} \lim_{x \to -\infty} f(x) &= K \end{align*}

if and only if for every \(\epsilon \gt 0\) there exists \(N \in \mathbb{R}\) so that \(|f(x)-K| \lt \epsilon\) whenever \(x \lt N\text{.}\)

Note that we can loosen the above requirement that \(f\) be defined on the whole real line — all we actually require is that it is defined all \(x\) larger than some value. It would be sufficient to require “there is some \(x_0 \in \mathbb{R}\) so that \(f\) is defined for all \(x \gt x_0\)”.

For completeness lets prove Theorem 1.5.3 using this form definition. The layout of the proof will be very similar to our proof of Theorem 1.4.1.