Properties of Limits

#calc/concept
If $L,M,c,$ and $k$ are real numbers and ${\displaystyle \underset{x\to c}{lim}f(x)=L}$ and ${\displaystyle \underset{x\to c}{lim}g(x)=M}$:

• Sum rule:
  • ${\displaystyle \underset{x\to c}{lim}(f(x)+g(x))=L+M}$
• Difference rule:
  • ${\displaystyle \underset{x\to c}{lim}(f(x)-g(x))=L-M}$
• Product rule:
  • ${\displaystyle \underset{x\to c}{lim}(f(x)\ast g(x))=L\ast M}$
• Constant Multiple rule:
  • ${\displaystyle \underset{x\to c}{lim}(k\ast f(x))=k\ast L}$
• Quotient rule:
  • ${\displaystyle \underset{x\to c}{lim}\left(\frac{f(x)}{g(x)}\right)=\frac{L}{M}}$ if $M\ne 0$
• Power rule:
  • ${\displaystyle \underset{x\to c}{lim}(f(x){)}^{\frac{r}{s}}=L^{\frac{r}{s}}}$