Definition:Limit of Real Function/Definition 1
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Definition
Let $\openint a b$ be an open real interval.
Let $c \in \openint a b$.
Let $f: \openint a b \setminus \set c \to \R$ be a real function.
Let $L \in \R$.
$\map f x$ tends to the limit $L$ as $x$ tends to $c$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$
where $\R_{>0}$ denotes the set of strictly positive real numbers.
Notation
$\map f x$ tends to the limit $L$ as $x$ tends to $c$, is denoted:
- $\map f x \to L$ as $x \to c$
or
- $\ds \lim_{x \mathop \to c} \map f x = L$
The latter is voiced:
- the limit of $\map f x$ as $x$ tends to $c$.
Sources
- 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions
- 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 $1.3.1$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 8.3$
- For a video presentation of the contents of this page, visit the Khan Academy.