Intuitive idea

Getting a Feel for Limits

In this chapter, the goal is not to prove anything or use formal symbols, but to build an everyday, intuitive sense of what a limit is. Later chapters on limits and continuity will turn these ideas into precise definitions and rules. Here, we are just learning to “see” limits.

Approaching a Value vs. Reaching a Value

Imagine walking toward a wall. You start 10 meters away and keep moving closer: 5 meters away, 2 meters away, 1 meter away, 0.5 meters away, and so on. You are approaching the wall. Whether or not you actually touch the wall at some moment is a different question.

A limit is about this “approach” idea:

• We look at what values a function gets close to when the input gets close to some number.
• We do not ask what the function equals exactly at that input.

In symbols, when you later see something like
$$\lim_{x \to a} f(x) = L,$$
you should read it as:
“as $x$ gets closer and closer to $a$, the values $f(x)$ get closer and closer to $L$.”

The key intuition:

• A limit is a statement about nearby behavior, not necessarily about what happens at the point.

A Simple Table Example

Consider the function
$$f(x) = x^2.$$

Look at what happens as $x$ gets close to $2$ from values near $2$:

• If $x = 1.9$, then $f(x) = 1.9^2 = 3.61$
• If $x = 1.99$, then $f(x) = 1.99^2 = 3.9601$
• If $x = 1.999$, then $f(x) = 1.999^2 = 3.996001$
• If $x = 2.1$, then $f(x) = 2.1^2 = 4.41$
• If $x = 2.01$, then $f(x) = 2.01^2 = 4.0401$
• If $x = 2.001$, then $f(x) = 2.001^2 = 4.004001$

As $x$ gets closer to $2$, $f(x)$ gets closer to $4$. From both sides of $2$, the values of $f(x)$ “home in” on $4$.

Even without symbols, you feel:
“Near $x=2$, $x^2$ is very close to $4$.”

So we say informally:
$$\lim_{x \to 2} x^2 = 4.$$

The important part: we looked at values near $2$, not only at $x = 2$ exactly.

A Limit Can Exist Even If the Function Is “Broken” at the Point

Now consider a function that is artificially “broken” at $x = 2$:

$$
g(x) =
\begin{cases}
x^2 & \text{if } x \neq 2,\\
100 & \text{if } x = 2.
\end{cases}
$$

So $g(x)$ is exactly like $x^2$, except at $x = 2$ it suddenly jumps to $100$.

Let’s look near $x = 2$ again:

• If $x = 1.9$, then $g(x) = 1.9^2 = 3.61$
• If $x = 1.99$, then $g(x) = 1.99^2 = 3.9601$
• If $x = 1.999$, then $g(x) = 1.999^2 = 3.996001$
• If $x = 2.1$, then $g(x) = 2.1^2 = 4.41$
• If $x = 2.01$, then $g(x) = 2.01^2 = 4.0401$
• If $x = 2.001$, then $g(x) = 2.001^2 = 4.004001$

All of these values are still very close to $4$. The fact that $g(2) = 100$ does not change the behavior near $2$.

So we still say, intuitively:
$$\lim_{x \to 2} g(x) = 4.$$

This shows a crucial idea:

• The limit looks at what happens as you approach $x = 2$.
• It does not care what the function does exactly at $x = 2$.

You can imagine a graph with a smooth curve approaching a certain height, but at that exact $x$ there might be a “hole” (no point) or a point placed somewhere else. The limit cares about the surrounding curve, not the “hole” or stray point.

Limits from the Left and from the Right

Sometimes it matters how you approach the point: from the left side (smaller values) or from the right side (larger values).

Intuitively:

• Approaching from the left of $a$ means using values like $a - 0.1$, $a - 0.01$, $a - 0.001$, and so on.
• Approaching from the right of $a$ means using values like $a + 0.1$, $a + 0.01$, $a + 0.001$, and so on.

We’ll later write these as left-hand and right-hand limits, but the basic picture is:

• If both sides are squeezing the function values toward the same number, that number is the limit.
• If the two sides are heading toward different numbers, the limit (from both sides together) does not exist.

You can picture walking toward a street corner from two different streets: if everyone ends up at the same corner, there’s a single meeting place; if people end up at different corners, there is no single “meeting point.”

When a Limit Does Not Exist (Intuitively)

There are several intuitive ways a limit can fail to exist. You do not need formal definitions here, just the feel.

1. Jump Behavior

Imagine a function whose graph looks like a step:

• As $x$ approaches $1$ from the left, the values get close to $2$.
• As $x$ approaches $1$ from the right, the values get close to $5$.

So the left side wants the limit to be $2$, the right side wants it to be $5$. There is a “jump” in the graph.

In this case, there is no single number that the function is approaching from both sides, so we say informally “the limit at $x = 1$ does not exist.”

2. Unbounded Growth (Blowing Up)

Consider the function
$$h(x) = \frac{1}{x^2}.$$

As $x$ approaches $0$ (from either side), the values $h(x)$ become larger and larger:

• $x = 0.1$ gives $h(x) = 100$
• $x = 0.01$ gives $h(x) = 10{,}000$
• $x = 0.001$ gives $h(x) = 1{,}000{,}000$

The function does not settle toward any finite number. Instead, it “blows up” without bound.

We will later talk about this using precise phrases, but intuitively: there is no finite limit at $x = 0$.

3. Endless Oscillation

Imagine a function that just keeps bouncing between values as you zoom in, never settling down. The classic example involves trigonometric functions, but the intuitive picture is enough:

• No matter how close you get to a certain $x$-value, the function keeps jumping back and forth between different outputs.
• There is no single “target” value that the outputs approach.

Again, there is no limit, because the values never settle near any one number.

Limits as “Predictions”

Another way to think about a limit is as a prediction based on nearby values.

Suppose you observe that for inputs:

• $x = 0.9$, the output is about $3.1$
• $x = 0.99$, the output is about $3.14$
• $x = 0.999$, the output is about $3.141$

Even if you don’t know what happens exactly at $x = 1$, you might predict: “If I could plug in $x = 1$, I expect the output would be $3.14159\ldots$ (something like $\pi$).”

A limit formalizes this idea:

• Even if you cannot evaluate the function at $x = 1$ (maybe it is undefined there),
• You can still say what number the outputs are trying to be as you get closer and closer to $1$.

So, intuitively, the limit is like the best guess for the function’s value at a point, based purely on its behavior extremely close to that point.

Why Limits Matter for Calculus

You do not need calculus yet, but it helps to know why we care about limits at all.

Many core ideas of calculus depend on limits:

• The slope of a curve at a point is defined as a limit of slopes of secant lines (average slopes over smaller and smaller intervals).
• The area under a curve is defined as a limit of areas of many thin rectangles.
• Many important functions are defined using limits when simpler definitions fail.

So having a solid intuitive sense of limits is like understanding what it means to “zoom in infinitely close” on a graph or a number. Later, the formal definitions in limits and continuity will make all of this precise.

Summary of the Intuitive Picture

Keep these core ideas in mind:

• A limit looks at what a function’s values approach as the input gets close to some number.
• The limit does not depend on what the function is exactly equal to at that number, and the function might not even be defined there.
• Approaching from the left and from the right should lead to the same value for the (two-sided) limit to exist.
• Limits can fail to exist if:
• The left and right approaches head toward different numbers (a jump),
• The function grows without bound near the point,
• Or the function oscillates endlessly without settling.

These ideas are the “feel” of limits. Later chapters will turn this intuition into precise definitions and techniques for computing limits.