Hereby L is the candidate value of limits, $\epsilon > 0$ is our degree of precision.
Introduce N as a "knob" for tuning $x \geq N$ to achieve the level of precision $\epsilon > 0$
N hereby is defined as when $x \geq N$, $|f(x) - L| \leq \epsilon$. (falls into $[L + \epsilon,L - \epsilon]$)
Trivially, as $\epsilon$ decreases, N has to get bigger.
Definition of Finite Limit at infinity: Suppose $f: (0,\infty) \rightarrow \mathbb{R}$ and $L \in \mathbb{R}$, then
$$
\lim\limits_{x \rightarrow \infty} = L
$$
if $\exists N > 0$ for which when $x > N$. $\forall \epsilon > 0$ there exists $|f(x)-L| \leq \epsilon$
Definition of Limit towards infinity: if $\forall k >0$, there exists $N < \infty$ such that when $x > N$, $f(x) > K$, then
$$
\lim\limits_{x \rightarrow \infty} = \infty
$$
Date: 2026-04-10
Hereby L is the candidate value of limits, $\epsilon > 0$ is our degree of precision.
Introduce N as a "knob" for tuning $x \geq N$ to achieve the level of precision $\epsilon > 0$
N hereby is defined as when $x \geq N$, $|f(x) - L| \leq \epsilon$. (falls into $[L + \epsilon,L - \epsilon]$)
Trivially, as $\epsilon$ decreases, N has to get bigger.
Definition of Finite Limit at infinity: Suppose $f: (0,\infty) \rightarrow \mathbb{R}$ and $L \in \mathbb{R}$, then
$$
\lim\limits_{x \rightarrow \infty} = L
$$
if $\exists N > 0$ for which when $x > N$. $\forall \epsilon > 0$ there exists $|f(x)-L| \leq \epsilon$
Definition of Limit towards infinity: if $\forall k >0$, there exists $N < \infty$ such that when $x > N$, $f(x) > K$, then
$$
\lim\limits_{x \rightarrow \infty} = \infty
$$