The Concept of a Limit

Concept	Quick Reference
Notation	$\underset{x\to a}{lim}f(x)=L$
Meaning	$f(x)$ gets arbitrarily close to $L$ as $x\to a$ (but $x\ne a$)
Key point	The value $f(a)$ does not affect the limit
Equivalent notation	$f(x)\to L$ as $x\to a$

Motivating Example

Consider the function $f$ defined by

$$f(x)=\frac{2x^2-2}{x-1}.$$

The function $f$ is defined for all values of $x$ except $x=1$, because substituting $x=1$ gives the undefined fraction $\frac{0}{0}$. But since $2x^2-2=2(x-1)(x+1)$, for $x\ne 1$ we can simplify:

f(x) = \frac{2\cancel{(x-1)}(x+1)}{\cancel{x-1}} = 2x + 2 \qquad (x \neq 1).

So the graph of $f$ is the line $y=2x+2$ with the single point $(1, 4)$ removed, shown as a hole in the following figure.

Now investigate the values of $f$ when $x$ is close to $1$ but not equal to $1$. Let $x$ take on values $0.9,0.95,0.99,0.999,0.9999,\dots$ or $1.1,1.05,1.01,1.001,1.0001,\dots$ The corresponding values of $f$ are shown in the following table.

From this table and the graph, we see that as $x$ gets closer and closer to $1$ (on either side), but never equals $1$, the value $f(x)$ gets closer and closer to $4$. More specifically, we can make $f(x)$ as close to $4$ as we desire by taking $x$ close enough to $1$. We express this by saying:

"The limit of $f(x)$ as $x$ approaches $1$ is $4$,"

and write

$$\underset{x\to 1}{lim}f(x)=4\text{or}f(x)\to 4\text{as}x\to 1.$$

The Intuitive Definition of a Limit

The Limit Is Independent of the Function Value at $a$

According to the definition, $x$ approaches $a$ but $x\ne a$, so the existence or non-existence of $f(a)$, and its value if it exists, has no bearing on the limit $\underset{x\to a}{lim}f(x)$.

For example, define a new function $g$ by

g(x) = \begin{cases} \dfrac{2x^2 - 2}{x - 1} & \text{if } x \neq 1 \\ 3 & \text{if } x = 1 \end{cases}

Then $g(x)$ and $f(x)={\displaystyle \frac{2x^2-2}{x-1}}$ agree everywhere except at $x=1$. Because the limit only depends on values near $1$ (not at $1$), we still have

$$\underset{x\to 1}{lim}g(x)=4,$$

even though $g(1)=3\ne 4$.

Frequently Asked Questions

How do I read the notation $\underset{x\to a}{lim}f(x)=L$?

Read it as: "the limit of $f(x)$ as $x$ approaches $a$ equals $L$." The arrow in $x\to a$ means $x$ is getting closer and closer to $a$, but never actually equals $a$.

Can a limit exist if the function is not defined at the point?

Yes. The limit $\underset{x\to a}{lim}f(x)$ depends only on the behavior of $f$ *near* $a$, not *at* $a$. The function $f(x)={\displaystyle \frac{2x^2-2}{x-1}}$ is not defined at $x=1$, yet $\underset{x\to 1}{lim}f(x)=4$.

How do I estimate a limit from a table of values?

Choose values of $x$ approaching $a$ from both the left and the right, getting steadily closer. If the values of $f(x)$ in both columns appear to stabilize toward the same number $L$, that number is the limit. However, tables can be misleading; algebraic confirmation is always better when possible.