Function/R/Limit/Epsilon/Rules/Fact

Rules for limits (function)

Let ${}T\subseteq \mathbb {R}$ denote a subset and ${}a\in \mathbb {R}$ a point. Let ${}f\colon T\rightarrow \mathbb {R}$ and ${}g\colon T\rightarrow \mathbb {R}$ denote functions, such that the limits ${}\operatorname {lim} _{x\rightarrow a}\,f(x)$ and ${}\operatorname {lim} _{x\rightarrow a}\,g(x)$ exist. Then the following statements hold.

1. The sum ${}f+g$ has in ${}a$ the limit

   ${}\operatorname {lim} _{x\rightarrow a}\,(f(x)+g(x))=\operatorname {lim} _{x\rightarrow a}\,f(x)+\operatorname {lim} _{x\rightarrow a}\,g(x)\,.$

2. The product ${}f\cdot g$ has in ${}a$ the limit

   ${}\operatorname {lim} _{x\rightarrow a}\,(f(x)\cdot g(x))=\operatorname {lim} _{x\rightarrow a}\,f(x)\cdot \operatorname {lim} _{x\rightarrow a}\,g(x)\,.$

3. Suppose that ${}g(x)\neq 0$ for all ${}x\in T$ and ${}\operatorname {lim} _{x\rightarrow a}\,g(x)\neq 0$. Then the quotient ${}f/g$ has in ${}a$ the limit

   ${}\operatorname {lim} _{x\rightarrow a}\,{\frac {f(x)}{g(x)}}={\frac {\operatorname {lim} _{x\rightarrow a}\,f(x)}{\operatorname {lim} _{x\rightarrow a}\,g(x)}}\,.$

Proof, Write another proof