Everything I Know About Limits

In the following definitions, let $p \in \R$ be a point and $f(x) : \R \to \R$ be a function defined on some open interval left to be specified. Where appropriate, $x_0 \in \R$ is the endpoint of an interval on which $f$ is defined.

Limits at a Point

Let $f$ be defined on some open interval around, but not necessarily including $p$. We say that $L \in \R$ is the limit of $f$ as $x$ approaches $p$ if for any $\epsilon > 0$ there exists some $\delta > 0$ so that for any $x \neq p$ and $\abs{x - p} < \delta$ we have $\abs{f(x) - L} < \epsilon$.

One-Sided Limits

Let $f$ be defined on some open interval $(x_0, p)$. Then we say that $R \in \R$ is the limit of $f$ as $x$ approaches $p$ from the left (or from below) if for any $\epsilon > 0$ there exists some $\delta > 0$ so that for any $x < p$ and $p - x < \delta$ we have $\abs{f(x) - R} < \epsilon$.

Let $f$ be defined on some open interval $(p, x_0)$. Then we say that $L \in \R$ is the limit of $f$ as $x$ approaches $p$ from the right (or from above) if for any $\epsilon > 0$ there exists some $\delta > 0$ so that for any $x > p$ and $x - p < \delta$ we have $\abs{f(x) - L} < \epsilon$.

Limits at Infinity

Let $f$ be defined on some interval $(x_0, \infty)$. Then we say that $L \in \R$ is the imit of $f$ as $x$ approaches $\infty$ if for any $\epsilon > 0$ there is some $x_\epsilon \in \R$ such that for all $x > x_\epsilon$ we have $\abs{f(x) - L} < \epsilon$

Let $f$ be defined on some interval $(-\infty, x_0)$. Then we say that $L \in \R$ is the limit of $f$ as $x$ approaches $-\infty$ if for any $\epsilon > 0$ there is some $x_\epsilon \in \R$ such that for all $x < x_\epsilon$ we have $\abs{f(x) - L} < \epsilon$

Let $c \in \R$ be a constant, $p \in \R \cup \set{\infty, -\infty}$ be a point, and $f, g : \R \to \R$ be functions such that $\lim_{x \to p} f(x) = L_f$ and $\lim_{x \to p} g(x) = L_g$. It is true that: