Limit is multiplicative: Difference between revisions

Revision as of 01:45, 16 October 2011

Statement

Suppose $f$ and $g$ are functions of one variable. Suppose $c\in \mathbb {R}$ is such that both $f$ and $g$ are defined on the immediate left and the immediate right of $c$ . Further, suppose that the limits $\lim _{x\to c}f(x)$ and $\lim _{x\to c}g(x)$ both exist (as finite numbers). In that case, the limit of the pointwise product of functions $f\cdot g$ exists and is the product of the individual limits:

$\lim _{x\to c}(f\cdot g)(x)=\lim _{x\to c}f(x)\cdot \lim _{x\to c}g(x)$

Equivalenty:

$\lim _{x\to c}[f(x)g(x)]=\lim _{x\to c}f(x)\cdot \lim _{x\to c}g(x)$

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 ==Statement==
-Suppose <math>f</math> and <math>g</math> are [[function]]s of one variable. Suppose <math>c \in \R</math> is such that both <math>f</math> and <math>g</math> are defined on the immediate left and the immediate right of <math>c</math>. Further, suppose that the [[fact about::limit]]s <math>\lim_{x \to c} f(x)</math> and <math>\lim_{x \to c} g(x)</math> both exist (as finite numbers). In that case, the limit of the [[fact about::pointwise product of functions]] <math>f \cdot g</math> exists and is the sum of the individual limits:
+Suppose <math>f</math> and <math>g</math> are [[function]]s of one variable. Suppose <math>c \in \R</math> is such that both <math>f</math> and <math>g</math> are defined on the immediate left and the immediate right of <math>c</math>. Further, suppose that the [[fact about::limit]]s <math>\lim_{x \to c} f(x)</math> and <math>\lim_{x \to c} g(x)</math> both exist (as finite numbers). In that case, the limit of the [[fact about::pointwise product of functions]] <math>f \cdot g</math> exists and is the product of the individual limits:
 <math>\lim_{x \to c} (f \cdot g)(x) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)</math>