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
    limx→a−f(x)=L
    limx→a+f(x)=L
  • Intuitive Definition of the Limit
    limx→af(x)=L

Glossary

infinite limit
A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a
intuitive definition of the limit
If all values of the function f(x) approach the real number L as the values of x(≠a) approach a, f(x) approaches L
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 x=a if the limit as x approaches a from the right or left is infinite