Limit: Difference between revisions

Revision as of 20:45, 20 October 2011

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).

@@ Line 35: / Line 35: @@
 The '''left hand limit''' (acronym '''LHL''') <math>\lim_{x \to c^-} f(x)</math> is defined as a value <math>L \in \R</math> such that <math>\lim_{x \to c^-} f(x) = L</math>. By the [[uniqueness theorem for limits]] (one-sided version), there is at most one value of <math>L \in \R</math> for which <math>\lim_{x \to c^-} f(x) = L</matH>. Hence, it makes sense to talk of ''the'' left hand limit when it exists.
-===Left hand limit===
+===Right hand limit===
 Suppose <math>f</math> is a [[function]] of one variable and <math>c \in \R</math> is a point such that <math>f</math> is defined to the immediate right of <math>c</math> (note that <math>f</math> may or may not be defined at <math>c</math>). In other words, there exists some value <math>t  > 0</math> such that <math>f</math> is defined on <math>(c,c+t)</math>.