Limit of quotient equals quotient of limits: Difference between revisions

Latest revision as of 01:49, 16 October 2011

Statement

Suppose $f$ and $g$ are functions of one variable. Suppose $c \in R$ 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) and that $lim_{x \to c} g (x) \neq 0$ . In that case, the limit of the pointwise quotient of functions $f / g$ exists at $c$ and is the quotient of the individual limits:

$lim_{x \to c} (\frac{f}{g}) (x) = \frac{lim_{x \to c} f (x)}{lim_{x \to c} g (x)}$

Equivalenty:

$lim_{x \to c} \frac{f (x)}{g (x)} = \frac{lim_{x \to c} f (x)}{lim_{x \to c} g (x)}$

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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) ''and'' that <math>\lim_{x \to c} g(x) \ne 0</math>. In that case, the limit of the [[fact about::pointwise quotient of functions]] <math>f/g</math> exists and is the quotient 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) ''and'' that <math>\lim_{x \to c} g(x) \ne 0</math>. In that case, the limit of the [[fact about::pointwise quotient of functions]] <math>f/g</math> exists at <math>c</math> and is the quotient of the individual limits:
 <math>\lim_{x \to c} \left(\frac{f}{g}\right)(x) = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}</math>