Summary of the Limit of a Function | Calculus I

Module 2: Limits

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
$\underset{x\to a^-}{\lim}f(x)=L$
$\underset{x\to a^+}{\lim}f(x)=L$
Intuitive Definition of the Limit
$\underset{x\to a}{\lim}f(x)=L$

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 $x(\ne a)$ 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