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

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.