Summary of the Precise Definition of a Limit

Essential Concepts

  • The intuitive notion of a limit may be converted into a rigorous mathematical definition known as the epsilon-delta definition of the limit.
  • The epsilon-delta definition may be used to prove statements about limits.
  • The epsilon-delta definition of a limit may be used to find deltas algebraically.
epsilon-delta definition of the limit
[latex]\underset{x\to a}{\lim}f(x)=L[/latex] if for every [latex]\varepsilon >0[/latex], there exists a [latex]\delta >0[/latex] such that if [latex]0<|x-a|<\delta[/latex], then [latex]|f(x)-L|<\varepsilon[/latex]
triangle inequality
If [latex]a[/latex] and [latex]b[/latex] are any real numbers, then [latex]|a+b|\le |a|+|b|[/latex]