Limit is multiplicative: Difference between revisions
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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 sum of the individual limits: | ||
<math>\lim_{x \to c} (f \cdot g)(x) = \lim_{x \to c} f(x) \cdot \ | <math>\lim_{x \to c} (f \cdot g)(x) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)</math> | ||
Equivalenty: | Equivalenty: | ||
<math>\lim_{x \to c}[f(x)g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c}g(x)</math> | <math>\lim_{x \to c}[f(x)g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c}g(x)</math> | ||
Revision as of 01:41, 16 October 2011
Statement
Suppose and are functions of one variable. Suppose is such that both and are defined on the immediate left and the immediate right of . Further, suppose that the limits and both exist (as finite numbers). In that case, the limit of the pointwise product of functions exists and is the sum of the individual limits:
Equivalenty: