# Lesson Series

Function limits show how close a function can get to a point on the y-axis as it approaches a number on the x-axis. Here we introduce notation and begin developing intuition through pictures.

There are two important scenarios for which a function limit does not exist. The first is when the limit produces two different values dependent upon whether we approach from the left or the right. The second is when the function approaches positive or negative infinity.

Here we provide the mathematically rigorous definition of a function limit and provide explicit steps on how to prove a function limit exists.

We provide a mathematically rigorous definition of one-sided limits and prove a very important relationship between two-sided and one-sided limits.

We list here important properties of function limits.

We define both an intuition and a rigorous mathematical definition of continuous functions.

This lesson provides a list of continuous functions.

We provide some useful examples and a graph that summarizes what we have learned about function limits.

We highlight what we have learned with regard to function limits of quotients of functions whose limits go to zero.

We discuss the precise definition of the function limit
\begin{equation}
\lim_{x\to\infty}\,f(x)=L.
\end{equation}
When this limit exists the number $L$ is called a horizontal asymptote.

We summarize the properties of continuous functions.

In this lesson we discuss a subtle point not needed by non-mathematics majors. Because we need to know the value of $f(x)$ in the domain $(a-\delta,a)\cup (a,a+\delta)$ to prove
\[
\lim_{x\to a}\,f(x)=L,
\]
the number $a$ must be inside the domain and not just in it. We define the difference between in and inside.

In this lesson we expand the idea of continuity to be more than at a single point. A function that is uniformly continuous over some interval is continuous at every point in that interval. We also show that continuous functions are uniformly continuous.