Calculus Vol. 1 · Section LetterChapter 2 · Limits

The opening problem

The Limit of a Function

Where a function is headed near a point — even when the point itself cannot be trusted, or plugged in at all.

Figure 2.2.1: $f(x)=\dfrac{x^2-4}{x-2}$ — the line $y=x+2$ with the point $(2,4)$ punched out.

Masthead in the annual-letter grammar: letterhead strip, an accent-barred kicker, a weight-500 serif title with a key word in accent, an italic lede — and the section's live opening figure to the right.

Calculus Vol. 1 · Section Letter§2.2 · The Limit of a Function

Dear reader,

This section is about a single, patient question: where is a function headed as its input creeps toward a point? Sometimes the point is a smooth part of the curve. Sometimes there is a hole, a jump, or a wall. The limit is the language that describes all four at once.

Read for the destination the graph is approaching, not the value it may or may not hold at the point itself.

In this section you will

IUsing correct notation, describe the limit of a function

IIUse a table of values to estimate a limit — or see when it fails

IIIUse a graph to estimate a limit — or see when it fails

IVDefine one-sided limits, with examples

VRelate one-sided and two-sided limits

VIUsing correct notation, describe an infinite limit

VIIDefine a vertical asymptote

The letter opens on drop-cap prose at the measure; the seven objectives sit in a numbered contents column that reveals one row at a time.

Calculus Vol. 1 · Section LetterThe opening problem

Closing in on

x = 2

A value the formula can't give you

For $f(x)=\dfrac{x^2-4}{x-2}$ , the point $x=2$ has no value — you would divide by zero. So where is the graph headed as $x$ draws close? Pick $x$ -values just below and just above 2, and watch $f(x)$ .

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

Table of values — both sides of 2

from left	$f(x)$	from right	$f(x)$
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

The opening problem in prose + a compact data table; the table and the conclusion each reveal on click.

Calculus Vol. 1 · Section LetterFrom one case to a method

Name the method

Do it again — for any function

You found where $f$ was heading as $x \to 2$ — without ever using $x = 2$ itself. The point of mathematics is to write down exactly what you did, so it works for the next function too:

1Choose

x

-values closing in from the left.

2And matching values from the right.

3Compute

f(x)

at each one.

4When both columns close in on one number, that is where

f

is heading.

The method, named

The Closing-In Method

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

The classroom "name your method" step, rendered as a numbered reveal plus an inline data card that states and defines the Closing-In Method.

Calculus Vol. 1 · Section LetterThe question driving §2.2

The question driving this section

How can we describe where a function is headed near a point — even when we can't, or shouldn't, plug the point in?

The hole at $x = 2$ is only the first way a point can't be trusted. By the end you will have a name for every way a function can behave as $x$ approaches — and every way that behavior can fail.

The section's driving question set as a large italic pull-quote — the annual-letter substitute for a cover slide.

Calculus Vol. 1 · Section LetterI · What a limit is

Definition 2.2.1 — Limit (intuitive)

What the neighborhood is doing

Let $f(x)$ be defined for all $x$ in an open interval containing $a$ , with the possible exception of $a$ itself, and let $L$ be a real number. If all values of $f(x)$ approach $L$ as the values of $x\ (\ne a)$ approach $a$ , then the limit of $f(x)$ as $x$ approaches $a$ is $L$ .

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

The notation

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

Open interval containing a, with a shaded band around a

Figure 2.2.2: the shaded band around $a$ is the neighborhood the limit reads — the point $a$ itself may be excluded.

The intuitive definition as drop-cap prose at the measure; the formal notation and the neighborhood figure sit in a ruled box on the right.

Calculus Vol. 1 · Section LetterThe big idea

Why the hole?

Headed, not there

Factor and cancel:

\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. The cancellation is legal only for $x \ne 2$ — which is 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.

Factor-and-cancel shows the hole; the boxed "Big Idea" states the section's thesis and reveals on click.

Calculus Vol. 1 · Section LetterII · Estimating from a table

Example 2.2.1 — a limit from a table

The value of $\dfrac{\sin x}{x}$ at $0$

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

$x$	$\sin x / x$
$\pm 0.1$	0.99833
$\pm 0.01$	0.99998
$\pm 0.001$	0.9999998

Both sides march toward 1: $\displaystyle\lim_{x \to 0}\frac{\sin x}{x}=1.$

Graph of sin(x)/x approaching 1 at x=0 with a hole

Figure 2.2.3: $\dfrac{\sin x}{x}$ heads to 1 at $x=0$ , where the function itself is undefined.

Estimating a limit from a table, figure-right; the table and the conclusion reveal on click.

Calculus Vol. 1 · Section LetterYour turn

Try It Now 2.2.1

Estimate from a table

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

Simplify first: $\dfrac{\frac{1}{x}-1}{x-1}=\dfrac{-(x-1)}{x(x-1)}=-\dfrac{1}{x}$ for $x \ne 1$ .

Both columns march toward $-1$ , so $\displaystyle\lim_{x \to 1}\frac{\frac{1}{x}-1}{x-1}=-1.$ Answer: $-1$ .

Table of values — both sides of 1

$x$	$f(x)$	$x$	$f(x)$
0.9	−1.1111	1.1	−0.9091
0.99	−1.0101	1.01	−0.9901
0.999	−1.0010	1.001	−0.9990

A commit-first try-it: prompt shows, then the simplification, table, and answer reveal in turn.

Calculus Vol. 1 · Section LetterIII · Estimating from a graph

Example 2.2.3 — a limit from a graph

When the point and the limit disagree

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

The marked point shows $g(-1)=4$ , but the curve approaches height 3 from both sides: $\displaystyle\lim_{x \to -1} g(x)=3.$

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

Graph of g(x) with limit 3 at x=-1 but a marked point at height 4

Figure 2.2.4: the curve heads to 3 at $x=-1$ , while the plotted point sits at 4.

Reading a limit off a graph, figure-right; the limit value and the "may disagree" caveat reveal on click.

Calculus Vol. 1 · Section LetterInterlude

From estimates to building blocks

Guesswork, and what comes next

Tables and graphs are quick, but they rest on guesswork — you are reading a trend, not proving a value. The next section builds algebraic limit laws that compute limits exactly.

Before we leave the informal stage, two limits are so basic that every one of those laws will lean on them.

A short bridge paragraph in the letter voice, pointing forward to the limit laws of §2.3.

Calculus Vol. 1 · Section LetterThe atomic limits

Theorem 2.2.1 — two important limits

The building blocks of every limit law

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.

For any real $a$ and constant $c$

\lim_{x \to a} x = a \qquad \lim_{x \to a} c = c.

The two atomic limits stated in a ruled figure box beside a one-paragraph gloss.

Calculus Vol. 1 · Section LetterHow a limit misses

Two ways a limit can fail

Oscillation, or blow-up

For a two-sided limit to exist, $f(x)$ must approach a single real number from both sides. It can miss in two very different ways — and the rest of the section gives each its own precise vocabulary.

Way one

Oscillation

$f$ bounces around forever without settling.

Way two

Blow-up

$f$ shoots off toward $\pm\infty$ — a vertical wall.

Both get labeled DNE for now — but each is about to earn its own precise name.

The two failure modes as two inline data cards; the "DNE for now" line reveals on click.

Calculus Vol. 1 · Section LetterOscillation

Example 2.2.4 — a limit that fails

The wandering $\sin(1/x)$

Evaluate $\displaystyle\lim_{x \to 0}\sin\!\left(\frac{1}{x}\right)$ . As $x \to 0$ , the input $1/x$ blows up, so $\sin(1/x)$ oscillates faster and faster — never settling.

Along $x=\dfrac{2}{\pi},\dfrac{2}{3\pi},\dfrac{2}{5\pi},\dots$ the value cycles $1,-1,1,-1,\dots$ forever. Different paths to 0 give different values, so $\displaystyle\lim_{x \to 0}\sin\!\left(\frac{1}{x}\right)$ DNE.

Graph of sin(1/x) oscillating without bound near x=0

Figure 2.2.5: $\sin(1/x)$ oscillates ever faster near $0$ — no single value to head toward.

Oscillation as a genuine DNE, figure-right; the multiple-paths argument reveals on click.

Calculus Vol. 1 · Section LetterIV · Toward one-sided limits

When "DNE" hides the real story

A failure worth describing

Saying "the limit does not exist" sometimes throws away useful information. $\sin(1/x)$ genuinely wandered — nothing more to say.

But recall $g(x)=\dfrac{|x-2|}{x-2}$ : it settles cleanly on $-1$ from the left and $+1$ from the right.

The two-sided limit fails, yet each side is perfectly well-behaved. That deserves its own notation.

Figure 2.2.6: $g(x)=\dfrac{|x-2|}{x-2}$ — clean on each side, a jump at $x=2$ .

Motivating one-sided limits with the live step-function figure; the "deserves its own notation" line reveals on click.

Calculus Vol. 1 · Section LetterIV · One-sided limits

Definition 2.2.2 — one-sided limits

Approaching from one side

If $f(x)$ approaches $L$ as $x\ (x < a)$ approaches $a$ from the left, we write $\displaystyle\lim_{x \to a^-} f(x) = L$ . If it approaches $L$ as $x\ (x > a)$ approaches $a$ from the right, we write $\displaystyle\lim_{x \to a^+} f(x) = L$ .

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 $\displaystyle\lim_{x \to 2^+} g(x) = 1$ .

From the left · from the right

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

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

Superscript sign: $a^-$ reads "just below $a$ "; $a^+$ reads "just above $a$ ".

One-sided limits as drop-cap prose; the two notations sit in a ruled box on the right.

Calculus Vol. 1 · Section LetterEvaluating side limits

Example 2.2.5 — evaluating one-sided limits

A jump at $x = 2$

Take the piecewise function $f(x)=\begin{cases} x+1 & x < 2,\\[2pt] x^2-4 & x \ge 2.\end{cases}$

Use the correct branch on each side: $\displaystyle\lim_{x \to 2^-} f(x) = 3$ and $\displaystyle\lim_{x \to 2^+} f(x) = 0$ .

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

Graph of the piecewise function jumping at x=2

Figure 2.2.7: the branches meet at different heights — a jump discontinuity at $x=2$ .

Evaluating both side limits of a piecewise function, figure-right; the values and the DNE verdict reveal on click.

Calculus Vol. 1 · Section LetterYour turn

Try It Now 2.2.4

Side limits of $\dfrac{|x^2-4|}{x-2}$

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

Factor: $|x^2-4| = |x-2|\,|x+2|$ . For $x > 2$ , the ratio $\to |x+2| = 4$ ; for $x < 2$ , it $\to -|x+2| = -4$ .

The two side limits

\lim_{x \to 2^-}\frac{|x^2-4|}{x-2}=-4

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

Answer: left side $-4$ ; right side $+4$ ; the two-sided limit DNE.

A try-it on side limits; the factoring reveals first, then the boxed answer.

Calculus Vol. 1 · Section LetterV · The handshake

Theorem 2.2.2 — one-sided and two-sided

When the two sides shake hands

The relationship

\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.

The iff relating side limits to the two-sided limit, boxed; the plain-language gloss reveals on click.

Calculus Vol. 1 · Section LetterVI · The vertical wall

Back to the blow-up

A function that never settles

Every function so far stayed bounded near its trouble point. But the third opening function never settles:

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

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

Figure 2.2.8: $h(x)=\dfrac{1}{(x-2)^2}$ climbs without bound on both sides of $x=2$ .

Motivating infinite limits with the live blow-up figure and the display limit to

+\infty

.

Calculus Vol. 1 · Section LetterVI · Infinite limits

Definition 2.2.3 — infinite limits

Recording how a limit fails

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 $+\infty$ records how the limit fails — by blowing up — instead of just labeling it "DNE". (Infinity is a description of behavior, not a real-number value.)

Increase / decrease without bound

\lim_{x \to a} f(x) = +\infty

\lim_{x \to a} f(x) = -\infty

Infinite limits as drop-cap prose; the two signed-infinity notations sit in a ruled box.

Calculus Vol. 1 · Section LetterVII · Vertical asymptotes

Definition 2.2.4 — vertical asymptote

Naming the wall

The line $x = a$ is a vertical asymptote of $f(x)$ if any one of the following holds:

Any one is enough

\lim_{x \to a^-} f(x) = \pm\infty,\quad \lim_{x \to a^+} f(x) = \pm\infty,\quad \lim_{x \to a} f(x) = \pm\infty.

Example: $h(x)=\dfrac{1}{(x-2)^2}$ has $\displaystyle\lim_{x \to 2} h(x) = +\infty$ , so the line $x = 2$ is a vertical asymptote.

The vertical-asymptote definition, boxed; the worked instance reveals on click.

Calculus Vol. 1 · Section LetterRecognizing an infinite limit

Example 2.2.6 — an infinite limit

The two sides of $\dfrac{1}{x}$ at $0$

Evaluate the one- and two-sided limits of $\dfrac{1}{x}$ at $x = 0$ .

$x$	$1/x$	$x$	$1/x$
−0.01	−100	0.01	100
−0.001	−1000	0.001	1000

The sides blow up in opposite directions, so $\displaystyle\lim_{x \to 0^-}\frac{1}{x}=-\infty$ , $\displaystyle\lim_{x \to 0^+}\frac{1}{x}=+\infty$ , and the two-sided limit DNE.

Graph of 1/x with opposite infinite branches at x=0

Figure 2.2.9: $\dfrac{1}{x}$ dives to $-\infty$ on the left and climbs to $+\infty$ on the right of $0$ .

Opposite one-sided infinities for

1/x

, figure-right; the table and the verdict reveal on click.

Calculus Vol. 1 · Section LetterEven vs. odd powers

Theorem 2.2.3 — powers of

(x-a)

Even climbs, odd splits

If $n$ is even

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

Both sides climb together to $+\infty$ .

If $n$ is odd

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

The sides split to opposite infinities.

For a real number $a$ and a positive integer $n$ — either way, the line $x = a$ is a vertical asymptote.

The even/odd dichotomy in two side-by-side boxes; the "either way, a vertical asymptote" line reveals on click.

Calculus Vol. 1 · Section LetterFinding a vertical asymptote

Example 2.2.7 — finding an asymptote

The even case $\dfrac{1}{(x+3)^4}$

Evaluate the limits of $f(x)=\dfrac{1}{(x+3)^4}$ at $x = -3$ , and identify any vertical asymptote.

Match the template $\dfrac{1}{(x-a)^n}$ : here $a = -3$ and $n = 4$ (even).

Even $n$ means both sides shoot to $+\infty$ , and because the limit blows up at $x=-3$ , the line $x = -3$ is a vertical asymptote.

Both sides climb

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

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

Vertical asymptote: $x = -3$ .

Applying Theorem 2.2.3 in the even case; the template match reveals, then the boxed infinite limits.

Calculus Vol. 1 · Section LetterYour turn — the odd case

Try It Now 2.2.6

The odd case $\dfrac{1}{(x-2)^3}$

Evaluate the limits of $f(x)=\dfrac{1}{(x-2)^3}$ at $x = 2$ and identify any vertical asymptote.

$n = 3$ is odd, so the side limits go to opposite infinities. The sides disagree, so the two-sided limit DNE — but one side blowing up is enough.

Opposite infinities

\lim_{x \to 2^-}\frac{1}{(x-2)^3}=-\infty

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

Answer: left $-\infty$ ; right $+\infty$ ; two-sided DNE; vertical asymptote at $x = 2$ .

The odd-power counterpart to Example 2.2.7; the reasoning reveals, then the boxed answer.

Calculus Vol. 1 · Section LetterAll four behaviors

Example 2.2.8 — four behaviors on one graph

Reading every case at once

Read each behavior off the graph of $f$ :

$x = -4$ (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 $\to -\infty$ , $f(3)$ undefined.

One graph showing a smooth point, a hole, a jump, and a vertical asymptote

Figure 2.2.10: smooth at $-4$ , a hole at $-2$ , a jump at $1$ , an asymptote at $3$ .

All four limit behaviors read off a single graph, figure-right; each point reveals on click.

Calculus Vol. 1 · Section LetterThe result to carry forward

The question, answered

One graph, four verdicts

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

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.

The four behaviors

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 four-behaviors summary as a compact table beside the closing thesis; the thesis line reveals on click.

Calculus Vol. 1 · Section LetterFin

Read the neighborhood,
and name how the function arrives.

Table · graph · one-sided · infinite — every way a limit can hold or fail.

Yours in the limit,

— the Calculus Volume 1 reader

Next: §2.3 The Limit Laws — computing limits exactly, so the tables and graphs become proofs.

←→SPACENavigate & reveal

A signed farewell in serif italic with an accent signature line and the key-hint capsule; the next-section pointer reveals on the final click.