4.2 Definition of Limits

Suppose that there is a function that calculates the profit of a company if it sells $x$ units of product:

$\pi (x)=\frac{-x^3+320x^2-\mathrm{26,000}x+\mathrm{400,000}}{x-100}$

We want to find the profit when 100 units of product are sold. However, we cannot not find the exact profit because the function is not defined at $x=100$. To understand the behavior of this function, we can calculate the profit when the company sells 95, 99, 99.5, 99.9, 99.99, 100.01, 100.1, 100.5, 101, and 105 units. When we plug these units into the profit function, the profit is as follows:

While this profit function is not a conventional profit function since there is a hole when $x$ is equal to 100, based on the pattern shown in the table above, we would expect that if the company sold 100 units, the profit would be $8,000. This leads us to the definition of limits.

The expression

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

is read, “the limit of $f(x)$ as $x$ approaches $c$ is $L$.” This means that a function $f(x)$ takes values arbitrarily close to $L$ as $x$ becomes arbitrarily close to $a$. In other words, the value of $f(x)$ tends to get closer to $L$ as $x$ gets closer to $a$ from both sides of $a$ while $x\ne a$.

If a function $f(x)$ takes values arbitrarily close to $L$ as $x$ becomes arbitrarily close to $a$ from the left side of $a$ (meaning that $x<a$ always, but $x$ is getting closer to $a$), it is called a left-hand limit, denoted as follows:

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

If a function $f(x)$ takes values arbitrarily close to $L$ as $x$ becomes arbitrarily close to $a$ from the right side of $a$ (meaning that $x>a$ always, but $x$ is getting closer to $a$), it is called a right-hand limit, denoted as follows:

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

Given our first definition of a limit and the definition of a one-sided limit, the following is true:

$\underset{x\to a}{lim}f(x)=L\text{ if and only if}\underset{x\to a^{-}}{lim}f(x)=L\text{ and}\underset{x\to a^{+}}{lim}f(x)=L$

Here are some examples. What is the limit of the following function as $x$ approaches 1?

$f(x)=\{\begin{array}{ll}x+1& \text{if}x\le 1\\ \frac{x^2-1}{x-1}& \text{if}x>1\end{array}$

Let’s evaluate!

As the above table shows, from the left side of 1,

$\underset{x\to 1^{-}}{lim}f(x)=\underset{x\to 1^{-}}{lim}x+1=2$

and from the right side of 1,

$\underset{x\to 1^{+}}{lim}f(x)=\underset{x\to 1^{+}}{lim}\frac{x^2-1}{x-1}=2$

Therefore, there exists a limit as $x$ approaches 1 such that

$\underset{x\to 1}{lim}f(x)=2$

Let’s look at another example. For the following function, find the limit as $x$ approaches -1.

$f(x)=\{\begin{array}{ll}x+1& \text{if}x>-1\\ 5& \text{if}x=-1\\ \frac{x^2}{x-1}& \text{if}x<-1\end{array}$

We can check again from the left side and the right side:

$\underset{x\to -1^{-}}{lim}f(x)=\underset{x\to -1^{-}}{lim}x+1=0$

$\underset{x\to -1^{+}}{lim}f(x)=\underset{x\to -1^{+}}{lim}\frac{x^2}{x-1}=-\frac{1}{2}$

Since the left-hand limit is not equal to the right-hand limit, the limit does not exist as $x$ approaches -1. You may wonder why the answer is not 5. This is because 5 is the value of $f(x)$ when $x$ equals exactly -1. Since we are trying to figure out the behavior of the function as $x$ approaches $x$ (not -1 but extremely close to -1 from the right and left sides), 5 is not the answer.

Now, how about a function of a fraction? Let’s look at a simple example to learn a few things. Suppose there is a function such that

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

What happens to the function as $x$ approaches 0? Since $x$ is squared, we know that the denominator is always positive. As $x$ gets closer to 0, the function returns larger values since the denominator of the fraction gets smaller. (If you want to see how it changes as $x$ gets close to 0, you can use a calculator to calculate the outcomes with different inputs for $x$.) As $x$ approaches 0, the function does not converge to a certain number but rather explodes to a greater number, which we denote as follows:

$\underset{x\to 0^{-}}{lim}f(x)=\mathrm{\infty }$

$\underset{x\to 0^{+}}{lim}f(x)=\mathrm{\infty }$

By definition, we then have

$\underset{x\to 0}{lim}f(x)=\mathrm{\infty }$

This is read as “the limit of $f(x)$ is infinity as $x$ approaches 0.” Since we cannot have 0 in the denominator, we know that $f(x=0)$ is undefined. What happens to the function if $x$ becomes extremely large or small? We can plug a value for $x$ into the function, but the larger the independent variable $x$, the larger the denominator is, and thus the smaller the outcome of the function. We denote this behavior as follows:

$\underset{x\to -\mathrm{\infty }}{lim}f(x)=0$

$\underset{x\to +\mathrm{\infty }}{lim}f(x)=0$

Given these limits, we can roughly sketch the graph with some additional information. We know that the function is always positive since the denominator of the function is always positive, and a positive number divided by another positive number is positive.

Let’s make sure we have mastered our understanding of the limits of functions with another example. Suppose we have the following information:

$\underset{x\to -2^{-}}{lim}f(x)=1$

$f(-2)=-1$

$\underset{x\to -2^{+}}{lim}f(x)=1$

$\underset{x\to 1^{-}}{lim}f(x)=-3$

$f(1)=-3$

$\underset{x\to 3^{-}}{lim}f(x)=-4$

$f(3)=2$

$\underset{x\to 3^{+}}{lim}f(x)=3$

Can you determine if there exists a limit as $x$ approaches -2, 1, and 3? If the limit exists, what is the limit?

Let’s look at the function piece by piece. First, when $x$ approaches -2, since there are limits identified from the right side and left side of -2, we can identify whether the limit exists or not. Since the left limit and right limit as $x$ approaches -2 are the same, the limit exists as $x$ approaches -2, and it is equal to 1. How about as $x$ approaches 1? We cannot determine whether the limit exists or not since we do not have information on the right limit. Finally, as $x$ approaches 3, since there are both left and right limits, we can identify whether the limit exists. Since the left limit and right limit as $x$ approaches 3 are different, the limit of the function does not exist as $x$ approaches 3.

Let’s go over one more example. Suppose that we have the following graph of a function:

Does the limit exist as $x$ approaches $a$, $b$, $c$, $d$, and $e$?

The question we need to ask is whether the function approaches the same outcome as $x$ approaches those values. As $x$ approaches $a$, the function comes to the same value; therefore, the limit exists as $x$ approaches $a$. How about $b$? Even though the actual value returned by the function is not the same as the right and left limits, since the function approaches the same value as $x$ approaches $b$ from both sides, we know that the limit exists as $x$ approaches $b$. The same can be said for when $x$ approaches $d$ and $e$. Therefore, the limit exists as $x$ approaches $d$ and $e$. Finally, for $c$, the left limit and the right limit as $x$ approaches $c$ are different. Therefore, the limit does not exist as $x$ approaches $c$.