Summary of the Limit of a Function

Essential Concepts

  • A table of values or graph may be used to estimate a limit.
  • If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
  • If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.
  • We may use limits to describe infinite behavior of a function at a point.

Key Equations

  • One-Sided Limits
    [latex]\underset{x\to a^-}{\lim}f(x)=L[/latex]
    [latex]\underset{x\to a^+}{\lim}f(x)=L[/latex]
  • Intuitive Definition of the Limit
    [latex]\underset{x\to a}{\lim}f(x)=L[/latex]

Glossary

infinite limit
A function has an infinite limit at a point [latex]a[/latex] if it either increases or decreases without bound as it approaches [latex]a[/latex]
intuitive definition of the limit
If all values of the function [latex]f(x)[/latex] approach the real number [latex]L[/latex] as the values of [latex]x(\ne a)[/latex] approach [latex]a[/latex], [latex]f(x)[/latex] approaches [latex]L[/latex]
one-sided limit
A one-sided limit of a function is a limit taken from either the left or the right
vertical asymptote
A function has a vertical asymptote at [latex]x=a[/latex] if the limit as [latex]x[/latex] approaches [latex]a[/latex] from the right or left is infinite