Definition:Limit of Real Function

Limit from the Left
Let $f$ be a real function defined on an open interval $\left({a \, . \, . \, b}\right)$.

Suppose that:
 * $\exists L: \forall \epsilon > 0: \exists \delta > 0: b - \delta < x < b \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$.

That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every real number in the domain of $f$, less than $b$ but within $\delta$ of $b$, has an image within $\epsilon$ of some real number $L$.


 * LimitFromLeft.png

Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $b$ from the left, and we write:
 * $f \left({x}\right) \to L$ as $x \to b^-$

or
 * $\displaystyle \lim_{x \to b^-} f \left({x}\right) = L$

This is voiced:
 * the limit of $f \left({x}\right)$ as $x$ tends to $b$ from the left.

Sometimes the notation $\displaystyle f \left({b^-}\right) = \lim_{x \to b^-} f \left({x}\right)$ is seen.

Limit from the Right
Let $f$ be a real function defined on an open interval $\left({a \, . \, . \, b}\right)$.

Suppose that $\exists L: \forall \epsilon > 0: \exists \delta > 0: a < x < a + \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$.

That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every real number in the domain of $f$, greater than $a$ but within $\delta$ of $a$, has an image within $\epsilon$ of some real number $L$.


 * LimitFromRight.png

Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $a$ from the right, and we write:
 * $f \left({x}\right) \to L$ as $x \to a^+$

or
 * $\lim_{x \to a^+} f \left({x}\right) = L$

This is voiced
 * the limit of $f \left({x}\right)$ as $x$ tends to $a$ from the right.

Sometimes the notation $\displaystyle f \left({a^+}\right) = \lim_{x \to a^+} f \left({x}\right)$ is seen.

Limit
Let $f$ be a real function defined on an open interval $\left({a \, . \, . \, b}\right)$ except possibly at some $c \in \left({a \, . \, . \, b}\right)$.

Suppose that $\exists L: \forall \epsilon > 0: \exists \delta > 0: 0 < \left|{x - c}\right| < \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$.

That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every real number in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some real number $L$.


 * LimitOfFunction.png

Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $c$, and we write:
 * $f \left({x}\right) \to L$ as $x \to c$

or
 * $\displaystyle \lim_{x \to c} f \left({x}\right) = L$

This is voiced:
 * the limit of $f \left({x}\right)$ as $x$ tends to $c$.

It can directly be seen that this definition is the same as that for a general metric space.