Limit is multiplicative: Difference between revisions

Latest revision as of 01:52, 16 October 2011

Statement

Statement for two functions

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 at $c$ 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:

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

Statement for multiple functions

Suppose $f_{1},f_{2},\dots ,f_{n}$ are all functions defined on the immediate left and immediate right of a point $c\in \mathbb {R}$ , and that all the limits $\lim _{x\to c}f_{1}(x)$ , $\lim _{x\to c}f_{2}(x)$ , $\dots$ , $\lim _{x\to c}f_{n}(x)$ exist (as finite numbers). Then, the limit of the pointwise product of functions $f_{1}\cdot f_{2}\cdot \dots \cdot f_{n}$ exists at $c$ and is the product of the individual limits:

$\lim _{x\to c}(f_{1}\cdot f_{2}\cdot \dots f_{n})(x)=\lim _{x\to c}f_{1}(x)\lim _{x\to c}f_{2}(x)\dots \lim _{x\to c}f_{n}(x)$

One-sided versions

The statements above have valid one-sided versions. For the left versions, we need both the functions to be defined on the immediate left of $c$ . For the right versions, we need the functions to be defined on the immediate right of $c$ .

Each of the statements below is true whenever the right side expressions make sense:

$\lim _{x\to c^{-}}[f(x)g(x)]]\lim _{x\to c^{-}}f(x)\cdot \lim _{x\to c^{-}}g(x)$
$\lim _{x\to c^{+}}[f(x)g(x)]]\lim _{x\to c^{+}}f(x)\cdot \lim _{x\to c^{+}}g(x)$
$\lim _{x\to c^{-}}(f_{1}\cdot f_{2}\cdot \dots f_{n})(x)=\lim _{x\to c^{-}}f_{1}(x)\lim _{x\to c^{-}}f_{2}(x)\dots \lim _{x\to c^{-}}f_{n}(x)$
$\lim _{x\to c^{+}}(f_{1}\cdot f_{2}\cdot \dots f_{n})(x)=\lim _{x\to c^{+}}f_{1}(x)\lim _{x\to c^{+}}f_{2}(x)\dots \lim _{x\to c^{+}}f_{n}(x)$

@@ Line 1: / Line 1: @@
 ==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 product of the individual limits:
+===Statement for two functions===
+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  at <math>c</math> 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>
@@ Line 8: / Line 10: @@
 <math>\lim_{x \to c}[f(x)g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c}g(x)</math>
+===Statement for multiple functions===
+Suppose <math>f_1,f_2,\dots,f_n</math> are all functions defined on the immediate left and immediate right of a point <math>c \in \R</math>, and that all the limits <math>\lim_{x \to c} f_1(x)</math>, <math>\lim_{x \to c} f_2(x)</math>, <math>\dots</math>, <math>\lim_{x \to c} f_n(x)</math> exist (as finite numbers). Then, the limit of the pointwise product of functions <math>f_1 \cdot f_2 \cdot \dots \cdot f_n</math> exists at <math>c</math> and is the product of the individual limits:
+<math>\lim_{x \to c} (f_1 \cdot f_2 \cdot \dots f_n)(x) = \lim_{x \to c} f_1(x) \lim_{x \to c} f_2(x) \dots \lim_{x \to c} f_n(x)</math>
+===One-sided versions===
+The statements above have valid one-sided versions. For the left versions, we need both the functions to be defined on the immediate left of <math>c</math>. For the right versions, we need the functions to be defined on the immediate right of <math>c</math>.
+Each of the statements below is true whenever the right side expressions make sense:
+* <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>
+* <math>\lim_{x \to c^-} (f_1 \cdot f_2 \cdot \dots f_n)(x) = \lim_{x \to c^-} f_1(x) \lim_{x \to c^-} f_2(x) \dots \lim_{x \to c^-} f_n(x)</math>
+* <math>\lim_{x \to c^+} (f_1 \cdot f_2 \cdot \dots f_n)(x) = \lim_{x \to c^+} f_1(x) \lim_{x \to c^+} f_2(x) \dots \lim_{x \to c^+} f_n(x)</math>