Table of Contents
Intuitive idea of a limit
In earlier chapters you have worked with functions as input–output machines. Limits are about asking:
“What value is the function getting closer to when the input gets closer to a certain number?”
This question matters even if:
- The function is not defined at that input, or
- The function’s value at that input does not match the value it is “approaching” nearby.
We use limits to:
- Describe instantaneous speed and slopes of curves (derivatives, in the next chapter).
- Handle “holes” or jumps in graphs.
- Make precise what it means for a function to be continuous.
In this chapter we stay at the intuitive (but careful) level. The formal symbolic rules and proofs are the focus of the later “Formal limits” chapter.
Approaching, not reaching
When we say “$x$ approaches $a$,” we mean $x$ gets closer and closer to $a$ but we are not insisting that $x = a$.
We write:
$$\lim_{x \to a} f(x) = L$$
and read: “the limit of $f(x)$ as $x$ approaches $a$ equals $L$.”
The idea is:
- We look at values of $f(x)$ for $x$ very close to $a$ (both smaller and larger than $a$).
- If these $f(x)$ values get closer and closer to a single number $L$, then the limit exists and equals $L$.
The actual value $f(a)$, if it exists, is not what defines the limit. The limit is about nearby behavior, not the value at the point itself.
Limits from numerical tables
One way to “see” a limit is to compute function values for $x$ values close to $a$.
Example: Consider $f(x) = x^2$ and try to understand $\lim_{x \to 2} x^2$.
Take $x$ values approaching $2$ from both sides:
- From the left: $1.9,\ 1.99,\ 1.999$
- From the right: $2.1,\ 2.01,\ 2.001$
Compute:
- $1.9^2 = 3.61$
- $1.99^2 = 3.9601$
- $1.999^2 \approx 3.996001$
- $2.1^2 = 4.41$
- $2.01^2 = 4.0401$
- $2.001^2 \approx 4.004001$
As $x$ gets closer to $2$ from both sides, $x^2$ gets closer to $4$. We write:
$$\lim_{x \to 2} x^2 = 4.$$
Here $f(2) = 4$ as well, but that is not what makes the limit true; it simply happens to agree with the limit.
Limits from graphs
A graph gives a visual way to understand limits.
To estimate $\lim_{x \to a} f(x)$ from a graph:
- Trace along the curve from the left toward $x = a$.
- Trace along from the right toward $x = a$.
- If both traces approach the same height (the same $y$-value), that height is the limit.
It does not matter whether:
- The graph has a filled dot, an open circle, or no point at all at $x = a$.
- The function is defined there.
The limit only cares about how the graph behaves near $x = a$.
When limits exist (intuitive)
For a limit $\lim_{x \to a} f(x)$ to exist and equal $L$, we need:
- Approaching from the left:
- As $x$ values slightly less than $a$ move toward $a$, $f(x)$ values move toward $L$.
- Approaching from the right:
- As $x$ values slightly greater than $a$ move toward $a$, $f(x)$ values also move toward $L$.
If these two “sides” approach the same number $L$, the limit is $L$.
When they do not match, the limit does not exist at that point (we will often say “the two-sided limit does not exist”).
One-sided limits
Sometimes we want to look at only one direction of approach. This will be important later when we discuss jumps in functions and continuity.
We write:
- Left-hand limit:
$$\lim_{x \to a^-} f(x)$$
Only values of $x$ less than $a$ are considered. - Right-hand limit:
$$\lim_{x \to a^+} f(x)$$
Only values of $x$ greater than $a$ are considered.
Intuitively:
- $\lim_{x \to a^-} f(x)$ is the value $f(x)$ is approaching as $x$ creeps in toward $a$ from the left.
- $\lim_{x \to a^+} f(x)$ is the value $f(x)$ is approaching as $x$ creeps in toward $a$ from the right.
A key relationship:
- The (two-sided) limit $\lim_{x \to a} f(x)$ exists and equals $L$ if and only if:
- $\lim_{x \to a^-} f(x) = L$ and
- $\lim_{x \to a^+} f(x) = L$.
If the left-hand and right-hand limits are different, then the overall limit at $a$ does not exist.
Basic limit behaviors (without proofs)
In later chapters you will learn formal rules for computing limits. Here we record some informal behaviors that match our intuitive understanding.
Limits of “nice” functions
For many familiar functions (polynomials, and many simple combinations of them), nothing strange happens at ordinary points. For such a function $f$ and any real number $a$ where $f$ is well-behaved, we usually have:
$$\lim_{x \to a} f(x) = f(a).$$
Informally, if the function has no hole, jump, or blow-up at $a$, then you can find the limit just by direct substitution.
Example:
- $f(x) = 3x^2 - x + 1$ at $x = 1$.
- Directly substituting:
$$f(1) = 3(1)^2 - 1 + 1 = 3.$$ - Since polynomials are well-behaved everywhere:
$$\lim_{x \to 1} (3x^2 - x + 1) = 3.$$
We will later learn how to recognize and prove these results, but for now you can use this as a working rule for “nice” points of “nice” functions.
Indeterminate form $0/0$ (just as a warning)
When you try to find $\lim_{x \to a} f(x)$ for a fraction and both numerator and denominator go to $0$, you get an expression of the form $0/0$. This does not directly tell you the limit; more work is needed.
For example:
$$f(x) = \frac{x^2 - 1}{x - 1}.$$
At $x = 1$:
- Numerator: $1^2 - 1 = 0$.
- Denominator: $1 - 1 = 0$.
The expression $\frac{0}{0}$ is not a number. However, for $x \neq 1$,
$$\frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1.$$
For $x$ close to $1$ (but not equal to $1$), $f(x)$ behaves like $x + 1$, which is close to $2$. So:
$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2.$$
Note that:
- The limit exists and equals $2$.
- But the function is not defined at $x = 1$ itself (there is a “hole” there).
Continuity: an intuitive picture
Continuity is about functions that do not have breaks, jumps, or holes. Informally, a function is continuous at a point if:
“You can draw its graph at that point without lifting your pencil.”
More precisely (but still informally), a function $f$ is continuous at $x = a$ if all three of these conditions fit together:
- The limit as you approach $a$ exists:
$$\lim_{x \to a} f(x) \text{ exists},$$ - The function has a value at $a$:
$$f(a) \text{ is defined},$$ - The limit and the value agree:
$$\lim_{x \to a} f(x) = f(a).$$
If any of these fails, $f$ is not continuous at $a$.
Types of discontinuities (intuitive)
A point where a function is not continuous is called a discontinuity. The main types you will encounter:
Removable discontinuity (hole)
- The limit $\lim_{x \to a} f(x)$ exists.
- But either $f(a)$ is not defined, or it is defined but not equal to the limit.
In a graph, this appears as a “hole” in the curve at $x = a$. Sometimes the hole might have a point placed elsewhere at the same $x$-value.
Example idea:
- Graph looks like a straight line, except at one $x$-value where the point is missing (open circle).
- The nearby values suggest a limit, but the actual function skips that value.
Such discontinuities are called “removable” because we could “fix” continuity just by redefining $f(a)$ to equal the limit.
Jump discontinuity
- The left-hand limit and right-hand limit are both finite, but they are different:
$$\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x).$$
The graph literally “jumps” up or down when crossing $x = a$. You might see a filled dot on one side and an open circle at a different height on the other side.
This often happens in piecewise-defined functions, where the rule changes suddenly at some point.
Infinite or essential-type discontinuity
Here the function grows without bound as $x$ approaches $a$:
- As $x \to a$, $f(x)$ becomes huge in the positive or negative direction.
On the graph, this often shows up as a vertical asymptote: the curve shoots up or down near $x = a$ rather than approaching a finite limiting height.
In such a case we say the limit does not exist because there is no finite number that $f(x)$ is approaching.
Continuity on intervals
So far we have talked about continuity at a single point. We say a function is “continuous on an interval” (for example, continuous on $[a,b]$) if it is continuous at every point of that interval (with appropriate one-sided reasoning at the endpoints).
Intuitively:
- A function continuous on an interval has a graph you can draw over that entire stretch without lifting your pencil or encountering an infinite spike.
- Common functions like polynomials, exponentials, and sines/cosines are continuous everywhere on the real line.
Many powerful theorems in calculus (for example, results about maximum and minimum values and about solving equations) apply to functions that are continuous on intervals.
Why limits and continuity matter for calculus
Limits supply the language and tools to:
- Define the derivative precisely as a limit of average rates of change.
- Define the definite integral as a limit of sums (in later chapters).
- Analyze where functions behave nicely (are continuous) and where they fail.
Continuity, in turn, is the natural condition that guarantees limits and function values match. This makes calculus work smoothly on most of the functions we care about.
Later, in the “Formal limits” chapter, you will see exact symbolic definitions and rules. Here, the goal is to have a strong mental picture:
- Limits describe approaching behavior.
- Continuity says the function’s actual value fits what its nearby behavior suggests.