Understanding Limits: The Foundation

Before we can talk about the instantaneous rate of change, which is what a derivative represents, we need a way to talk about what happens to a function's value as its input gets extremely close to a certain point, without necessarily reaching it. This concept is the limit. Think of it as zooming in infinitely close to a point on a graph and observing where the function is heading.

Consider a function $f(x)$. We want to understand the behavior of $f(x)$ as $x$ approaches some value $a$. We're interested in the value that $f(x)$ gets closer and closer to as $x$ gets closer and closer to $a$, from either side (values less than $a$ and values greater than $a$), but not necessarily equal to $a$ itself.

Formalizing the Idea: The Limit

Mathematically, we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write this as:

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

This statement means that we can make the value of $f(x)$ arbitrarily close to $L$ by choosing $x$ sufficiently close to $a$, but not equal to $a$.

The function doesn't even need to be defined at $x=a$ for the limit to exist. This is a significant point. The limit describes the trend around the point, not the value at the point.

Let's look at an example. Consider the function:

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

What happens as $x$ approaches 1? If we try to plug in $x=1$, we get $\frac{1^2 - 1}{1 - 1} = \frac{0}{0}$, which is undefined. However, we can simplify the function for $x \neq 1$:

$f(x) = \frac{(x-1)(x+1)}{x-1} = x+1 \quad (\text{for } x \neq 1)$

Now, as $x$ gets very close to 1 (like 0.9, 0.99, 0.999 or 1.1, 1.01, 1.001), the value of $f(x)$ gets very close to $1+1=2$. Even though $f(1)$ is undefined, the limit exists:

$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$

The function $f(x) = (x^2 - 1) / (x - 1)$ behaves exactly like $f(x) = x+1$, except at $x=1$, where it's undefined (indicated by the open circle). The limit as $x$ approaches 1 is 2, representing the y-value the function approaches near the hole.

One-Sided Limits

Sometimes, we might only care about what happens as $x$ approaches $a$ from one specific direction:

• Limit from the left: $x$ approaches $a$ through values less than $a$. Denoted as $\lim_{x \to a^-} f(x)$.
• Limit from the right: $x$ approaches $a$ through values greater than $a$. Denoted as $\lim_{x \to a^+} f(x)$.

For the overall limit $\lim_{x \to a} f(x)$ to exist and be equal to $L$, both the left-hand limit and the right-hand limit must exist and be equal to $L$.

Why Limits Matter for Derivatives

The concept of a limit is the bedrock upon which the definition of the derivative is built. As previewed in the chapter introduction, the derivative measures the instantaneous rate of change, essentially the slope of a function at a single point.

How can we find the slope at a single point? We start by finding the slope of a line connecting two points on the function's curve (a secant line). Then, we use the concept of a limit to move one of these points infinitely close to the other. The limit of the slope of these secant lines, as the distance between the points approaches zero, gives us the slope of the tangent line at that single point. This limiting process is precisely what defines the derivative, which we will formally define in the next section. Understanding limits allows us to make this transition from an average rate of change over an interval to an instantaneous rate of change at a point.