Limit is multiplicative

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:

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