Limit

Definition

Two-sided limit

Suppose $f$ is a function of one variable and $c \in R$ is a point such that $f$ is defined to the immediate left and immediate right of $c$ (note that $f$ may or may not be defined at $c$ ). In other words, there exists some value $t > 0$ such that $f$ is defined on $(c - t, c) \cup (c, c + t)$ .

For a given value $L \in R$ , we say that:

$lim_{x \to c} f (x) = L$

if the following holds (the single sentence is broken down into multiple points to make it clearer):

For every $ϵ > 0$
there exists $δ > 0$ such that
for all $x \in R$ satisfying $0 < | x - c | < δ$ (explicitly, $x \in (c - δ, c) \cup (c, c + δ)$ ),
we have $| f (x) - L | < ϵ$ (explicitly, $f (x) \in (L - ϵ, L + ϵ)$ ).

The limit (also called the two-sided limit) $lim_{x \to c} f (x)$ is defined as a value $L \in R$ such that $lim_{x \to c} f (x) = L$ . By the uniqueness theorem for limits, there is at most one value of $L \in R$ for which $lim_{x \to c} f (x) = L$ . Hence, it makes sense to talk of the limit when it exists.

Left hand limit

Suppose $f$ is a function of one variable and $c \in R$ is a point such that $f$ is defined to the immediate left of $c$ (note that $f$ may or may not be defined at $c$ ). In other words, there exists some value $t > 0$ such that $f$ is defined on $(c - t, c)$ .

For a given value $L \in R$ , we say that:

$lim_{x \to c^{-}} f (x) = L$

if the following holds (the single sentence is broken down into multiple points to make it clearer):

For every $ϵ > 0$
there exists $δ > 0$ such that
for all $x \in R$ satisfying $0 < c - x < δ$ (explicitly, $x \in (c - δ, c)$ ),
we have $| f (x) - L | < ϵ$ (explicitly, $f (x) \in (L - ϵ, L + ϵ)$ .

The left hand limit (acronym LHL) $lim_{x \to c^{-}} f (x)$ is defined as a value $L \in R$ such that $lim_{x \to c^{-}} f (x) = L$ . By the uniqueness theorem for limits (one-sided version), there is at most one value of $L \in R$ for which $lim_{x \to c^{-}} f (x) = L$ . Hence, it makes sense to talk of the left hand limit when it exists.

Right hand limit

Suppose $f$ is a function of one variable and $c \in R$ is a point such that $f$ is defined to the immediate right of $c$ (note that $f$ may or may not be defined at $c$ ). In other words, there exists some value $t > 0$ such that $f$ is defined on $(c, c + t)$ .

For a given value $L \in R$ , we say that:

$lim_{x \to c^{+}} f (x) = L$

if the following holds (the single sentence is broken down into multiple points to make it clearer):

For every $ϵ > 0$
there exists $δ > 0$ such that
for all $x \in R$ satisfying $0 < x - c < δ$ (explicitly, $x \in (c, c + δ)$ ),
we have $| f (x) - L | < ϵ$ (explicitly, $f (x) \in (L - ϵ, L + ϵ)$ .

The right hand limit (acronym RHL) $lim_{x \to c^{+}} f (x)$ is defined as a value $L \in R$ such that $lim_{x \to c^{+}} f (x) = L$ . By the uniqueness theorem for limits (one-sided version), there is at most one value of $L \in R$ for which $lim_{x \to c^{+}} f (x) = L$ . Hence, it makes sense to talk of the right hand limit when it exists.

Relation between the limit notions

The two-sided limit exists if and only if (both the left hand limit and right hand limit exist and they are equal to each other).

Definition of limit in terms of a game

The formal definitions of limit, as well as of one-sided limit, can be reframed in terms of a game. This is a special instance of an approach that turns any statement with existential and universal quantifiers into a game.

Two-sided limit

Consider the limit statement, with specified numerical values of $c$ and $L$ and a specified function $f$ :

$lim_{x \to c} f (x) = L$

The game is between two players, a Prover whose goal is to prove that the limit statement is true, and a Skeptic (also called a Verifier or sometimes a Disprover) whose goal is to show that the statement is false. The game has three moves:

First, the skeptic chooses $ϵ > 0$ , or equivalently, chooses the target interval $(L - ϵ, L + ϵ)$ .
Then, the prover chooses $δ > 0$ , or equivalently, chooses the interval $(c - δ, c + δ) ∖ {c}$ .
Then, the skeptic chooses a value $x$ satisfying $0 < | x - c | < δ$ , or equivalently, $x \in (c - δ, c + δ) ∖ {c}$ , which is the same as $(c - δ, c) \cup (c, c + δ)$ .

Now, if $| f (x) - L | < ϵ$ (i.e., $f (x) \in (L - ϵ, L + ϵ)$ ), the prover wins. If $| f (x) - L | \geq ϵ$ , the skeptic wins.

We say that the limit statement

$l i m_{x \to c} f (x) = L$

is true if the prover has a winning strategy for this game. The winning strategy for the prover basically constitutes a strategy to choose an appropriate $δ$ in terms of the $ϵ$ chosen by the skeptic. Thus, it is an expression of $δ$ as a function of $ϵ$ .

We say that the limit statement

$l i m_{x \to c} f (x) = L$

is false if the skeptic has a winning strategy for this game. the winning strategy for the skeptic involves a choice of $ϵ$ , and a strategy that chooses a value of $x$ (constrained in the specified interval) based on the prover's choice of $δ$ .