Definition:Limit of Real Function/Left
Definition
Let $\openint a b$ be an open real interval.
Let $f: \openint a b \to \R$ be a real function.
Let $L \in \R$.
Suppose that:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: b - \delta < x < b \implies \size {\map f x - L} < \epsilon$
where $\R_{>0}$ denotes the set of strictly positive real numbers.
That is, for every real strictly positive $\epsilon$ there exists a real strictly 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 $L$.
Then $\map f x$ is said to tend to the limit $L$ as $x$ tends to $b$ from the left, and we write:
- $\map f x \to L$ as $x \to b^-$
or
- $\ds \lim_{x \mathop \to b^-} \map f x = L$
This is voiced:
- the limit of $\map f x$ as $x$ tends to $b$ from the left
and such an $L$ is called:
- a limit from the left.
Also known as
A limit from the left is also seen referred to as a left-hand-limit.
Some sources prefer to use a more direct terminology and refer to a limit from below. However, this may be confusing if the function $\map f x$ is decreasing.
Notation
Notations that may be encountered for the limit from the left:
- $\ds \lim_{x \mathop \to b^-} \map f x$
- $\map f {b^-}$ or $\map f {b -}$
- $\map f {b - 0}$
- $\ds \lim_{x \mathop \uparrow b} \map f x$
- $\ds \lim_{x \mathop \nearrow b} \map f x$
Also see
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.3$: Limits of functions: Definition $1.3.3$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 8.1$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): below
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): left-hand limit
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): limit from the left and right