12.1 Finding limits: numerical and graphical approaches

Intuitively, we know what a limit    is. A car can go only so fast and no faster. A trash can might hold 33 gallons and no more. It is natural for measured amounts to have limits. What, for instance, is the limit to the height of a woman? The tallest woman on record was Jinlian Zeng from China, who was 8 ft 1 in. https://en.wikipedia.org/wiki/Human_height and http://en.wikipedia.org/wiki/List_of_tallest_people Is this the limit of the height to which women can grow? Perhaps not, but there is likely a limit that we might describe in inches if we were able to determine what it was.

To put it mathematically, the function whose input is a woman and whose output is a measured height in inches has a limit. In this section, we will examine numerical and graphical approaches to identifying limits.

Understanding limit notation

We have seen how a sequence    can have a limit, a value that the sequence of terms moves toward as the nu mber of terms increases. For example, the terms of the sequence

gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function $f(x)=L\text{,}$ then as the input $x$ gets closer and closer to $a,$ the output y -coordinate gets closer and closer to $L.$ We say that the output “approaches” $L.$

[link] provides a visual representation of the mathematical concept of limit. As the input value $x$ approaches $a,$ the output value $f\left(x\right)$ approaches $L.$

We write the equation of a limit as

This notation indicates that as $x$ approaches $a$ both from the left of $x=a$ and the right of $x=a,$ the output value approaches $L.$

Consider the function

We can factor the function as shown.

Notice that $x$ cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition, $x\ne 7,$ for the simplified function. We can represent the function graphically as shown in [link] .

What happens at $x=7$ is completely different from what happens at points close to $x=7$ on either side. The notation

indicates that as the input $x$ approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7.

What happens at $x=7?$ When $x=7,$ there is no corresponding output. We write this as

This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as

Notice that the limit of a function can exist even when $f(x)$ is not defined at $x=a.$ Much of our subsequent work will be determining limits of functions as $x$ nears $a,$ even though the output at $x=a$ does not exist.