# Function/R/Limit/Epsilon/Rules/Fact

Let ${\displaystyle {}T\subseteq \mathbb {R} }$ denote a subset and ${\displaystyle {}a\in \mathbb {R} }$ a point. Let ${\displaystyle {}f\colon T\rightarrow \mathbb {R} }$ and ${\displaystyle {}g\colon T\rightarrow \mathbb {R} }$ denote functions, such that the limits ${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,f(x)}$ and ${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,g(x)}$ exist. Then the following statements hold.
1. The sum ${\displaystyle {}f+g}$ has in ${\displaystyle {}a}$ the limit
${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,(f(x)+g(x))=\operatorname {lim} _{x\rightarrow a}\,f(x)+\operatorname {lim} _{x\rightarrow a}\,g(x)\,.}$
2. The product ${\displaystyle {}f\cdot g}$ has in ${\displaystyle {}a}$ the limit
${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,(f(x)\cdot g(x))=\operatorname {lim} _{x\rightarrow a}\,f(x)\cdot \operatorname {lim} _{x\rightarrow a}\,g(x)\,.}$
3. Suppose that ${\displaystyle {}g(x)\neq 0}$ for all ${\displaystyle {}x\in T}$ and ${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,g(x)\neq 0}$. Then the quotient ${\displaystyle {}f/g}$ has in ${\displaystyle {}a}$ the limit
${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,{\frac {f(x)}{g(x)}}={\frac {\operatorname {lim} _{x\rightarrow a}\,f(x)}{\operatorname {lim} _{x\rightarrow a}\,g(x)}}\,.}$