Limits

In earlier parts of the course, functions were mostly things you could plug numbers into without trouble: you gave an input, you got an output. Limits extend this idea to focus not just on what the function does at a particular input, but on how the function behaves near that input.

This chapter introduces limits in an intuitive, pre-calculus way. More formal and technical treatment will come later in calculus. Here we concentrate on:

• What a limit is trying to capture.
• How limits describe “approaching” a value.
• Why limits are needed (especially for problems involving “instantaneous” change, like slopes of tangents).
• Basic notation and simple examples.

The idea of “approaching”

Imagine you are walking toward a door. As time goes on, your distance $d$ from the door gets smaller: 3 meters, 2 meters, 1 meter, 0.5 meter, and so on. You can describe what $d$ is doing as you approach the time $t$ when you reach the door.

There are two different questions here:

1. What is $d$ when you actually reach the door?
2. What values does $d$ get closer and closer to as time $t$ gets closer and closer to that moment?

In this case, both answers are 0, because when you reach the door your distance is exactly 0, and as you approach that moment your distance gets closer and closer to 0.

Limits are about the second kind of question: what value a function is getting close to as the input gets close to some number, no matter what happens exactly at that number.

The crucial idea in a limit is the phrase “gets close to” or “approaches.” We will not, in this chapter, give the full formal definition; instead, we will work with a consistent, intuitive picture.

Limit notation

Suppose $f$ is a function and $a$ is a number. We write

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

and read this as:

• “The limit of $f(x)$ as $x$ approaches $a$ is $L$.”

This means:

• As $x$ takes values closer and closer to $a$ (but not necessarily equal to $a$), the values $f(x)$ get closer and closer to $L$.

It does not say anything (yet) about what happens when $x$ is exactly equal to $a$. The function might:

• Equal $L$ at $x = a$,
• Be something different at $x = a$,
• Not even be defined at $x = a$.

The limit is about the trend near $x = a$, not the exact value at $x = a$.

A simple example

Consider the function

$$
f(x) = x^2.
$$

What is $\lim_{x \to 2} x^2$?

If you plug in values close to 2, you get:

• $x = 1.9 \Rightarrow f(x) = 3.61$
• $x = 1.99 \Rightarrow f(x) = 3.9601$
• $x = 2.01 \Rightarrow f(x) = 4.0401$
• $x = 2.1 \Rightarrow f(x) = 4.41$

As $x$ gets closer to 2, $x^2$ gets closer to 4. In fact, we can see directly that

$$
\lim_{x \to 2} x^2 = 4.
$$

Here, the limit matches the actual value at $x=2$ since $f(2) = 4$ as well. But that will not always be the case.

Limits versus actual values

A key reason for introducing limits is to separate two ideas:

• The value of a function at a point: $f(a)$.
• The value the function is approaching near that point: $\lim_{x \to a} f(x)$.

These can be:

• The same, or
• Different, or
• One of them might not even exist (for example, $f(a)$ might be undefined while the limit exists).

A classic example: a “hole” in the graph

Define

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

When $x \neq 1$, you can simplify:

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

But at $x = 1$, the expression is $\frac{0}{0}$, which is undefined. So $g(1)$ is not defined.

Still, we can study what happens near $x = 1$ by looking at values like:

• $x = 0.9 \Rightarrow g(x) = 1.9$
• $x = 0.99 \Rightarrow g(x) = 1.99$
• $x = 1.01 \Rightarrow g(x) = 2.01$
• $x = 1.1 \Rightarrow g(x) = 2.1$

As $x$ gets closer to 1, $g(x)$ gets closer to 2. So we say

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

even though $g(1)$ does not exist.

Graphically, the graph of $g(x)$ looks like the line $y = x + 1$, but with a “hole” at the point $(1, 2)$. The limit $\lim_{x \to 1} g(x)$ is the $y$-value that the graph is “aiming at” as $x$ approaches 1, even though the graph doesn’t actually include that point.

This illustrates a general theme:

• Limits describe the behavior near a point, not necessarily at the point.

Why limits matter

Limits are central because they allow you to:

• Talk about slopes of tangents to curves (instantaneous rates of change).
• Define the derivative of a function.
• Define the area under a curve precisely (through integration, later on).
• Make sense of expressions that look like $\frac{0}{0}$ in a careful way.
• Analyze functions that have jumps, holes, or other “strange” behavior.

You will see formal definitions and systematic rules in later calculus chapters; here we focus on recognizing the idea and reading limit notation correctly.

A “slope at a point” idea

You may already know that the slope of a straight line is “rise over run” between any two points. For a curved graph, the slope changes from point to point, so “slope at a point” is not obvious.

The idea is:

• Take two points on the curve very close to each other.
• Compute the slope of the line between them (a “secant” line).
• Move those two points closer and closer together.
• The slope of the secant line approaches a number: that limit is the slope of the tangent at the point (the derivative).

Formally, this uses a limit of a difference quotient. For now, you just need to see that limits capture what happens when we let one point slide closer and closer to another.

Approaching from two sides

When we say $\lim_{x \to a} f(x)$, we are usually allowing $x$ to approach $a$ from both smaller and larger values.

Sometimes, you want to focus on only one side:

• From the left (values less than $a$),
• Or from the right (values greater than $a$).

We use special notation for this:

• Left-hand limit:
  $$
  \lim_{x \to a^-} f(x)
  $$
  means “the limit of $f(x)$ as $x$ approaches $a$ from the left (with $x < a$).”
• Right-hand limit:
  $$
  \lim_{x \to a^+} f(x)
  $$
  means “the limit of $f(x)$ as $x$ approaches $a$ from the right (with $x > a$).”

The (two-sided) limit $\lim_{x \to a} f(x)$ exists and equals $L$ only if both:

• $\lim_{x \to a^-} f(x) = L$, and
• $\lim_{x \to a^+} f(x) = L$.

If the left and right limits are different, the overall limit does not exist.

Example where left and right differ

Consider

$$
h(x) =
\begin{cases}
1, & x < 0, \\
2, & x \ge 0.
\end{cases}
$$

At $x = 0$:

• From the left ($x < 0$): $h(x) = 1$, so $\lim_{x \to 0^-} h(x) = 1$.
• From the right ($x > 0$): $h(x) = 2$, so $\lim_{x \to 0^+} h(x) = 2$.

Because $1 \ne 2$, the (two-sided) limit $\lim_{x \to 0} h(x)$ does not exist.

This type of behavior is called a “jump.” The graph would look like a horizontal line at height 1 to the left of 0, and a horizontal line at height 2 to the right, with a jump at $x=0$.

Ways to explore limits

At this level, you’ll mainly use three complementary approaches to understand limits:

1. Tables of values:
   Choose values of $x$ that get closer and closer to $a$, and see what $f(x)$ seems to be approaching.
2. Graphs:
   Look at the graph of $y = f(x)$ and see what $y$-value the graph approaches as $x$ approaches $a$ along the curve.
3. Algebraic simplification:
   Sometimes an expression can be simplified so that the limit can be found by direct substitution into a simpler function (as in the “hole” example above).

More advanced techniques and rules (like special limit laws and algebraic tricks for harder expressions) will appear in later chapters on formal limits.

When a limit does not exist (intuitively)

There are several ways a limit can fail to exist at a point $x=a$. Intuitively, this may happen when:

• The function “jumps” to a different value depending on side (as in the piecewise example).
• The function grows without bound (shoots off to infinity).
• The function oscillates wildly without settling down near a single number.

In this chapter, it is enough to recognize that:

• If there is no single number that the function values get closer to as $x$ gets closer to $a$, we say the limit at $a$ does not exist.

Summary

• A limit describes what value a function’s outputs are approaching as the inputs get closer and closer to some number.
• $\lim_{x \to a} f(x) = L$ means that when $x$ is near $a$, $f(x)$ is near $L$.
• The limit depends on behavior near $x = a$, not necessarily the function’s value at $x = a$ itself.
• One-sided limits allow us to look at behavior from the left or from the right.
• The (two-sided) limit exists only if the left-hand and right-hand limits match.
• Limits are the foundation for defining derivatives and integrals in calculus, making them a central idea in precalculus and beyond.