Limit of Monotone Real Function

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Theorem

Increasing Function

Let $f$ be a real function which is increasing and bounded above on the open interval $\left({a \,.\,.\, b}\right)$.

Let the supremum of $f$ on $\left({a \,.\,.\, b}\right)$ be $L$.


Then:

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

where $\displaystyle \lim_{x \mathop \to b^-} f \left({x}\right)$ is the limit of $f$ from the left at $b$.


Decreasing Function

Let $f$ be a real function which is decreasing and bounded below on the open interval $\left({a \,.\,.\, b}\right)$.

Let the infimum of $f$ on $\left({a \,.\,.\, b}\right)$ be $l$.


Then:

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

where $\displaystyle \lim_{x \to b^-} f \left({x}\right)$ is the limit of $f$ from the left at $b$.