Combination Theorem for Limits of Functions/Real/Product Rule
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Theorem
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
- $\ds \lim_{x \mathop \to c} \map f x = l$
- $\ds \lim_{x \mathop \to c} \map g x = m$
Then:
- $\ds \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$
Proof
Let $\sequence {x_n}$ be a sequence of elements of $S$ such that:
- $\forall n \in \N: x_n \ne c$
- $\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
- $\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
- $\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Product Rule for Real Sequences:
- $\ds \lim_{n \mathop \to \infty} \paren {\map f {x_n} \map g {x_n} } = l m$
Applying Limit of Real Function by Convergent Sequences again, we get:
- $\ds \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$
$\blacksquare$
Sources
- 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions: Theorem $1 \ \text{(ii)}$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.4$: Continuity
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 8.12 \ \text{(ii)}$