2.2 The Limit of a Function

In this section, you will learn to:

Describe the limit of a function using correct notation (Definition 2.2.1).
Estimate a limit from a table of values — or see it fail (Examples 2.2.1, 2.2.6).
Estimate a limit from a graph — or see it fail (Examples 2.2.3, 2.2.8).
Define one-sided limits and give examples (Definition 2.2.2).
Relate one-sided limits to the two-sided limit (Theorem 2.2.2).
Describe an infinite limit using correct notation (Definition 2.2.3).
Define a vertical asymptote (Definition 2.2.4).

Here is the graph of $f(x) = \dfrac{x^2 - 4}{x - 2}$. At $x = 2$ it has no value — you'd divide by zero. So where is the graph headed as $x$ gets close to $2$? Pick $x$-values just below and just above $2$, and compute $f(x)$.

Figure 2.12 — A removable hole. The function $f(x)=(x^2-4)/(x-2)$ is undefined at $x=2$; two values on the curve crowd toward the missing point $f(2)$ from both sides.

**Table 2.2.1** — outputs of $f(x) = \frac{x^2 - 4}{x - 2}$ from both sides of $x = 2$.
from left, $x$	$f(x)$	from right, $x$	$f(x)$
1	3	3	5
1.5	3.5	2.5	4.5
1.9	3.9	2.1	4.1
1.99	3.99	2.01	4.01
1.999	3.999	2.001	4.001
1.9999	3.9999	2.0001	4.0001

From both sides the values of $f(x)$ close in on $4$ — even though $f(2) = \tfrac{0}{0}$ has no value. The graph heads toward $4$ without ever reaching it.

Before you read on

You just found the single value $f(x)$ was heading toward as $x$ got closer and closer to $2$ — without ever using $x = 2$ itself. Write down exactly what you did, step by step. Which $x$-values did you choose? What did you compute at each one? How did you decide on the final value, and how did you make sure both sides agreed?

The recipe, written out, is this: choose $x$-values closing in on the point from the left, and matching values from the right; compute $f(x)$ at each one; then read down both columns — when they close in on a single number, that's where $f$ is heading. Call it the Closing-In Method.

The Closing-In Method

To find the value a function is heading toward as $x$ gets close to a point, plug in $x$-values closer and closer from both sides and watch what number the outputs settle on.

The point of math is to do this again for any function, with language precise enough that two people always mean the same thing.

Definition 2.2.1: Intuitive Definition of a Limit

Let $f(x)$ be defined at every point in an open interval containing $a$, except possibly at $a$ itself, and let $L$ be a real number. If the values of $f(x)$ approach $L$ as the values of $x$ ($x \ne a$) approach $a$, then the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write

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

The limit reports what the neighborhood around $a$ is doing, not the value at the point itself. For $f(x) = \dfrac{x^2-4}{x-2}$, the value $f(2)$ does not exist — yet $\displaystyle\lim_{x \to 2} f(x) = 4$.

Factor and cancel to see why:

$$ \frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} = x+2 \quad \text{for } x \ne 2. $$

So $f$ is the line $y = x+2$ with the single point $(2, 4)$ punched out — a hole. The cancellation is only legal for $x \ne 2$, which is exactly why we approach the point instead of substituting it. (Evaluating limits exactly by factoring is the work of §2.3.)

The big idea

A limit reads where a function is headed, not where it is — like reading which way a friend is heading through a crowd, even when you cannot see exactly where they stand.

Example 2.2.1

Evaluate $\displaystyle\lim_{x \to 0}\frac{\sin x}{x}$ using a table of values. The function is even, so the left and right columns match.

Solution

**Table 2.2.2** — both sides march toward 1.
$x$	$\pm 0.1$	$\pm 0.01$	$\pm 0.001$
$\sin x / x$	0.99833	0.99998	0.9999998

Both sides march toward 1:

$$ \lim_{x \to 0}\frac{\sin x}{x} = 1. $$

Figure 2.13 — The graph of f(x) = sin(x)/x confirms the limit at x = 0 is 1, even though the function is undefined there.

Figure 2.13 — The graph of $f(x)=\sin x / x$ confirms the estimate from Table 2.2.2.

Try It Now 2.2.1

Estimate $\displaystyle\lim_{x \to 1}\frac{\frac{1}{x}-1}{x-1}$ from a table of values.

Solution

Simplify first, for $x \ne 1$:

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

**Table 2.2.3** — both columns march toward $-1$.
$x$	0.9	0.99	0.999	1.001	1.01	1.1
$f(x)$	−1.1111	−1.0101	−1.0010	−0.9990	−0.9901	−0.9091

Answer:

$$ \lim_{x \to 1}\frac{\frac{1}{x}-1}{x-1} = -1. $$

Example 2.2.3

For a function $g(x)$ shown on a graph, evaluate $\displaystyle\lim_{x \to -1} g(x)$.

Solution

The marked point on the graph shows $g(-1) = 4$, but the curve approaches height $3$ from both sides:

$$ \lim_{x \to -1} g(x) = 3. $$

The limit and the function value are allowed to disagree — the limit reports the neighborhood, not the isolated point.

$Figure 2.15 — The graph of g(x) where lim_{x -> -1} g(x) is the open-circle height but g(-1) = 4 (the marked dot above the curve).$

Figure 2.15 — The graph of $g(x)$ includes one point that does not lie on the smooth curve — it approaches height $3$, but $g(-1) = 4$.

Tables and graphs are quick, but they rest on guesswork — you are reading a trend, not proving a value. The next section (§2.3) builds algebraic limit laws that compute limits exactly. Before we leave this informal stage, though, two limits are so basic that every one of those laws will lean on them.

Theorem 2.2.1: Two Basic Limits

For any real number $a$ and constant $c$:

$$ \lim_{x \to a} x = a \qquad\text{and}\qquad \lim_{x \to a} c = c. $$

The identity function $f(x) = x$ marches along with $x$; a constant function never moves. These two limits are the atomic building blocks inside every limit law we prove from here on.

Two ways to fail

For a two-sided limit to exist, $f(x)$ must approach a single real number from both sides. Before reading on, can you think of two very different ways that could fail — two different ways the outputs could refuse to settle on one value?

There are two very different ways a limit can miss. In oscillation, $f$ bounces around forever without settling. In a blow-up, $f$ shoots off toward $\pm\infty$ — a vertical wall. Both get labeled DNE (does not exist) for now — but the rest of this section gives each its own precise vocabulary.

Example 2.2.4

Evaluate $\displaystyle\lim_{x \to 0}\sin\!\left(\frac{1}{x}\right)$.

Solution

As $x \to 0$, the input $1/x$ blows up, so $\sin(1/x)$ oscillates faster and faster — never settling on a value. Along $x = \frac{2}{\pi}, \frac{2}{3\pi}, \frac{2}{5\pi}, \dots$ the value cycles $1, -1, 1, -1, \dots$ forever. Different paths to $0$ give different values, so

$$ \lim_{x \to 0}\sin\!\left(\frac{1}{x}\right)\ \text{DNE.} $$

Figure 2.17 — The graph of $f(x)=\sin(1/x)$ oscillates rapidly between $-1$ and $1$ as $x$ approaches 0.

Saying "the limit does not exist" sometimes throws away useful information. $\sin(1/x)$ genuinely wandered — there's nothing more to say. But consider $g(x) = \dfrac{|x-2|}{x-2}$: it settles cleanly on $-1$ from the left and $+1$ from the right. The two-sided limit still fails, yet each side is perfectly well-behaved. That deserves its own notation.

Definition 2.2.2: One-Sided Limits

From the left. If $f(x)$ approaches $L$ as $x$ ($x < a$) approaches $a$:

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

From the right. If $f(x)$ approaches $L$ as $x$ ($x > a$) approaches $a$:

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

For $g(x) = \dfrac{|x-2|}{x-2}$: the two-sided limit DNE, but the side limits are clean — $\displaystyle\lim_{x \to 2^-} g(x) = -1$ and $\displaystyle\lim_{x \to 2^+} g(x) = 1$.

Example 2.2.5

Evaluate the one-sided limits of

$$ f(x) = \begin{cases} x+1 & \text{if } x < 2, \\ x^2-4 & \text{if } x \ge 2. \end{cases} $$

Solution

Use the correct branch on each side:

$$ \lim_{x \to 2^-} f(x) = 3, \qquad \lim_{x \to 2^+} f(x) = 0. $$

The side limits disagree, so $\displaystyle\lim_{x \to 2} f(x)$ DNE — the graph jumps at $x = 2$.

Figure 2.18 — The piecewise function with a jump at x = 2 between left-side height 3 and right-side height 0.

Figure 2.18 — The graph of the piecewise $f(x)$ has a break at $x=2$.

Try It Now 2.2.4

Find the one-sided limits of $\dfrac{|x^2-4|}{x-2}$ at $x = 2$.

Solution

Factor: $|x^2-4| = |x-2|\,|x+2|$. For $x > 2$:

$$ \frac{|x^2-4|}{x-2} \to |x+2| = 4. $$

For $x < 2$:

$$ \frac{|x^2-4|}{x-2} \to -|x+2| = -4. $$

$$ \lim_{x \to 2^-}\frac{|x^2-4|}{x-2} = -4, \qquad \lim_{x \to 2^+}\frac{|x^2-4|}{x-2} = 4. $$

Left side $-4$; right side $+4$; two-sided limit DNE.

Theorem 2.2.2: Relating One-Sided and Two-Sided Limits

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

The two-sided limit exists exactly when the two side limits shake hands on the same value. If they disagree, the two-sided limit does not exist.

Every function so far has stayed bounded near its trouble point. But a third kind of function never settles at all:

$$ h(x) = \frac{1}{(x-2)^2} \ \longrightarrow\ +\infty \quad \text{as } x \to 2. $$

This is the blow-up failure flagged a moment ago. Rather than write "DNE," we record how it fails — with the symbol $\infty$.

Definition 2.2.3: Infinite Limits

If $f(x)$ increases without bound as $x \to a$, we write $\displaystyle\lim_{x \to a} f(x) = +\infty$; if it decreases without bound, $\displaystyle\lim_{x \to a} f(x) = -\infty$. The same applies to the one-sided versions $x \to a^-$ and $x \to a^+$. Writing $\pm\infty$ records how the limit fails — infinity describes behavior, not a real-number value.

Definition 2.2.4: Vertical Asymptote

The line $x = a$ is a vertical asymptote of $f(x)$ if any one of $\displaystyle\lim_{x \to a^-} f(x) = \pm\infty$, $\displaystyle\lim_{x \to a^+} f(x) = \pm\infty$, or $\displaystyle\lim_{x \to a} f(x) = \pm\infty$ holds. For example, $h(x) = \dfrac{1}{(x-2)^2}$ has $\displaystyle\lim_{x \to 2} h(x) = +\infty$, so $x = 2$ is a vertical asymptote.

Example 2.2.6

Find the limits of $\dfrac{1}{x}$ at $x = 0$.

Solution

**Table 2.2.4** — the sides run off in opposite directions.
$x$	−0.01	−0.001	0.001	0.01
$1/x$	−100	−1000	1000	100

$$ \lim_{x \to 0^-}\frac{1}{x} = -\infty, \qquad \lim_{x \to 0^+}\frac{1}{x} = +\infty. $$

The sides blow up in opposite directions, so $\displaystyle\lim_{x \to 0}\frac{1}{x}$ DNE.

Figure 2.19 — The graph of f(x) = 1/x has opposite-side infinite limits at x = 0.

Figure 2.19 — The graph of $f(x)=1/x$ confirms that the limit as $x$ approaches 0 does not exist.

Theorem 2.2.3: Infinite Limits for Positive Integers $n$

If $n$ is even:

$$ \lim_{x \to a}\frac{1}{(x-a)^n} = +\infty. $$

If $n$ is odd:

$$ \lim_{x \to a^+}\frac{1}{(x-a)^n} = +\infty, \qquad \lim_{x \to a^-}\frac{1}{(x-a)^n} = -\infty. $$

Either way, the line $x = a$ is a vertical asymptote.

Example 2.2.7

Find the vertical asymptote of $f(x) = \dfrac{1}{(x+3)^4}$ at $x = -3$.

Solution

Match the template $\dfrac{1}{(x-a)^n}$: here $a = -3$ and $n = 4$ (even). Even $n$ means both sides shoot to $+\infty$:

$$ \lim_{x \to -3^-} f(x) = \lim_{x \to -3^+} f(x) = \lim_{x \to -3} f(x) = +\infty. $$

Because the limit blows up at $x = -3$, the line $x = -3$ is a vertical asymptote.

Try It Now 2.2.6

Find the one-sided limits of $f(x) = \dfrac{1}{(x-2)^3}$ at $x = 2$.

Solution

Here $n = 3$ is odd, so the side limits go to opposite infinities:

$$ \lim_{x \to 2^-}\frac{1}{(x-2)^3} = -\infty, \qquad \lim_{x \to 2^+}\frac{1}{(x-2)^3} = +\infty. $$

The sides disagree, so the two-sided limit DNE. But one side blowing up is enough: left $-\infty$; right $+\infty$; two-sided DNE; vertical asymptote at $x = 2$.

Example 2.2.8

Read four different behaviors off a single graph of $f$.

Solution

$x = -4$ (smooth): $\displaystyle\lim_{x \to -4} f(x) = 0$, and $f(-4) = 0$.
$x = -2$ (hole): $\displaystyle\lim_{x \to -2} f(x) = 3$, but $f(-2)$ is undefined.
$x = 1$ (jump): $\lim^- = 6$, $\lim^+ = 3$, two-sided DNE, $f(1) = 6$.
$x = 3$ (asymptote): both sides $\to -\infty$, $f(3)$ is undefined.

Where is a function headed near a point — even when the point itself can't be trusted? One graph can hold all four answers.

Figure 2.21 — A graph illustrating four kinds of behavior at four x-values: smooth zero (-4), removable hole (-2), jump (1), vertical asymptote (3).

Figure 2.21 — One graph, four kinds of behavior: smooth at $x=-4$, a hole at $x=-2$, a jump at $x=1$, and a vertical asymptote at $x=3$.

**Table 2.2.5** — the four verdicts, one per feature.
Point on the graph	Limit verdict
smooth ($x=-4$)	exists; $\lim = f(a)$
hole ($x=-2$)	exists; $\lim \ne f(a)$
jump ($x=1$)	two-sided DNE; sides clean
asymptote ($x=3$)	infinite; vertical asymptote

The limit always reads the neighborhood, never the point. When the neighborhood agrees, the limit exists; when it splits or blows up, we don't stop at "DNE" — we name how. Next: §2.3 The Limit Laws.

Point on the graph	Limit verdict
smooth (\(x=-4\))	exists; \(\lim = f(a)\)
hole (\(x=-2\))	exists; \(\lim \ne f(a)\)
jump (\(x=1\))	two-sided DNE; sides clean
asymptote (\(x=3\))	infinite; vertical asymptote