What is a limit?

A limit is the value that a function approaches when $x$ gets very close to some value $a\text{.}$ We are allowed to say

$\underset{x\to a}{\lim }f(x)=L$

if and only if we can make $f(x)$ as close to $L$ as we desire by making $x$ sufficiently close to $a\text{.}$ The value of $f(a)$ doesn’t matter—it could be different from $L$ or undefined for all we care. The purpose of the limit is to talk about what happens when $x$ is very close (but not equal) to $a\text{.}$

Suppose I want $f(x)$ to be no more than 0.00123 away from $L\text{.}$ If the limit exists, then this has to be possible by making $x$ a certain amount left or right of $a\text{.}$ The definition I gave above is informal, but you should be able to see that it is quite a bit more precise than simply saying, “it gets really close to $L\text{.}$”

When we say claim that a limit exists, we are implying that the function approaches the limit from both sides—left and right. If it doesn’t do this, the limit does not exist. However, we can still talk about one side by itself using different notation. The left and right limits are represented by

$\underset{x\to a^{-}}{\lim }f(x)$ and $\underset{x\to a^{+}}{\lim }f(x)\text{.}$

In some cases it is easy to determine a limit just by looking at a graph. Consider the limit as $x$ approaches 5 for the following function:

Notice that $f(5)=2\text{.}$ Remember, though, the actual value of the function when $x$ is equal to 5 is irrelevant to the limit as $x$ approaches 5. From the graph, you should be able to see that

$\underset{x\to 5^{-}}{\lim }f(x)=4$ and $\underset{x\to 5^{+}}{\lim }f(x)=1\text{,}$

and since these are not equal, $\underset{x\to 5}{\lim }f(x)$ does not exist.