Calculus Volume 1 · Chapter 2 · Limits

The Limit of a Function

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

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Titlepage in the Zhangzara grid grammar: italic-serif title on the graph-paper canvas, cobalt double rule, one-sentence lede, hairline rule, mono byline; the pixel-glitch scanline stair carries the decorative signature.

Using correct notation, describe the limit of a function definition 2.2.1
Use a table of values to estimate a limit — or see it fail example 2.2.1
Use a graph to estimate a limit — or see it fail example 2.2.3
Define one-sided limits and provide examples definition 2.2.2
Relate one-sided and two-sided limits theorem 2.2.2
Describe an infinite limit in correct notation definition 2.2.3
Define a vertical asymptote definition 2.2.4

A ledger-style outline: mono leading-zero numerals, italic-serif entries on hairline rules, a right-aligned tag column pointing at the object that delivers each objective. Each reveals on click.

This 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 $x$ closes in on 2? Tabulate $x$ -values just below and just above.

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

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

Figure 1: $f(x)=\dfrac{x^2-4}{x-2}$ — the line $y=x+2$ with the point $(2,4)$ punched out.

The opening problem in figure-right layout: prose and the closing-in ledger sit left, the live graph video sits right. Table then takeaway reveal on click; the last row is bold cobalt because it is the tightest estimate.

Definition 2.2.1 — Limit (intuitive)

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 real. If all values of $f(x)$ approach $L$ as the values of $x\ (\ne a)$ approach $a$ , then

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

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

Open interval containing a, with a shaded neighborhood band around a

Figure 2: An open interval containing $a$ — the shaded band is the neighborhood the limit reads.

Figure-right definition master: the ruled, cobalt-topped definition box and its neighborhood diagram. The takeaway — limit reads the neighborhood, not the point — reveals on click.

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 only legal 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.)

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.

The factoring that explains the hole, then the big idea in a ruled result box revealed on click: a limit reads where a function is headed, not where it is.

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:

$x$	$\sin x / x$
$\pm 0.1$	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 near x = 0

Figure 3: $y=\dfrac{\sin x}{x}$ — a removable hole at $x=0$ , the curve heading to height 1.

Estimating a limit from a table: prompt left, the even-function ledger and the verdict reveal on click, the graph confirms it on the right.

Estimate a limit

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 $x \ne 1$ .

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

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

Commit-first practice: prompt shows, then simplification, the ledger, and the answer reveal one click at a time. Answer −1.

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:

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

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

Graph of g(x) with an open circle at height 3 and a solid dot at (−1, 4)

Figure 4: $g(x)$ near $x=-1$ — the curve heads to 3 while $g(-1)=4$ .

Reading a limit off a graph: the curve heads to 3 even though the plotted point is 4 — limit and value may disagree. Reveals on click.

Tables and graphs rest on guesswork — you read a trend, not a proof. §2.3 builds algebraic limit laws that compute limits exactly; two limits are so basic that every law will lean on them.

Theorem 2.2.1 — Two important 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 are the atomic building blocks inside every limit law from here on.

The two atomic limits — identity and constant — stated in a ruled theorem box. They are the building blocks the §2.3 limit laws are proved from.

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.

Oscillation

$f$ bounces around forever without settling on any value.

Blow-up

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

Both get labeled DNE for now — but the rest of the section gives each its own precise vocabulary.

Two failure modes in ruled cards — oscillation and blow-up — with the footnote that both are DNE for now but each earns its own name later. Footnote reveals on click.

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 $1,-1,1,-1,\dots$ . Different paths give different values, so

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

Graph of sin(1/x) oscillating ever faster near x = 0

Figure 5: $y=\sin(1/x)$ — infinitely many oscillations as $x \to 0$ .

The oscillation failure: sin(1/x) cycles 1, −1 forever along shrinking inputs, so the limit does not exist. Reveals on click; graph on the right.

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

A DNE that hides a clean story: the sign function settles on −1 and +1 on its two sides. That motivates one-sided limits. Video on the right stays live.

Definition 2.2.2 — One-sided limits

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

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

From the right. If $f(x)$ approaches $L$ as $x\ (x \gt a)$ 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 $\displaystyle\lim_{x \to 2^+} g(x) = 1$ .

Left and right limits defined side by side in one ruled box; the sign-function reading — −1 on the left, +1 on the right — reveals as an abstract on click.

Evaluate the one-sided limits of

f(x)=\begin{cases} x+1 & \text{if } x \lt 2,\\[2pt] x^2-4 & \text{if } x \ge 2.\end{cases}

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 $x = 2$ .

Piecewise graph with a jump at x = 2, left branch to 3 and right branch to 0

Figure 7: A piecewise function jumping at $x=2$ : left $\to 3$ , right $\to 0$ .

Evaluating each one-sided limit off the correct branch: 3 from the left, 0 from the right, so the two-sided limit DNE. Reveals on click.

Estimate one-sided limits

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

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

For $x \gt 2$ : $\ \dfrac{|x^2-4|}{x-2} \to |x+2| = 4$ . For $x \lt 2$ : $\ \dfrac{|x^2-4|}{x-2} \to -|x+2| = -4$ .

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

Answer: left side $-4$ ; right side $+4$ (two-sided DNE).

Practice one-sided limits: factor the absolute value, read −4 on the left and +4 on the right, so the two-sided limit DNE. Reveals on click.

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.

The bridge theorem: a two-sided limit exists iff both one-sided limits exist and agree. Stated as an iff in a ruled box.

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 flagged earlier. Rather than write "DNE", we record how it fails — with the symbol $\infty$ .

Figure 8: $h(x)=\dfrac{1}{(x-2)^2}$ — both sides rise without bound at $x=2$ .

The blow-up case returns: 1/(x−2)² rises without bound at x = 2. Instead of DNE we record how it fails, with ∞. Video stays live on the right.

Definition 2.2.3 — Infinite limits

If $f(x)$ increases without bound as $x \to a$ , write $\displaystyle\lim_{x \to a} f(x) = +\infty$ ; if it decreases without bound, $\displaystyle\lim_{x \to a} f(x) = -\infty$ . Same for $x \to a^-$ and $x \to a^+$ .

$+\infty$ records how the limit fails — infinity describes behavior, it is not a real value.

Definition 2.2.4 — Vertical asymptote

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

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

e.g. $h(x)=\dfrac{1}{(x-2)^2}$ has $\lim_{x \to 2} h = +\infty$ , so $x = 2$ is a vertical asymptote.

Two definitions in paired ruled boxes: infinite limits record how a limit blows up, and any one infinite (one- or two-sided) limit makes x = a a vertical asymptote.

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

\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.

Graph of 1/x with a vertical asymptote at x = 0

Figure 9: $y=\dfrac{1}{x}$ — $-\infty$ from the left, $+\infty$ from the right of 0.

The two sides of 1/x blow up in opposite directions at 0, so the two-sided limit DNE even though each side is infinite. Ledger and verdict reveal on click.

Theorem 2.2.3 — Infinite limits from positive integers

For a real number $a$ and a positive integer $n$ :

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

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

The parity theorem for reciprocal powers: even n sends both sides to +∞; odd n splits them to ±∞. Either way x = a is a vertical asymptote.

Find the vertical asymptote

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 $+\infty$ :

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

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

Applying Theorem 2.2.3: n = 4 is even, so both sides of 1/(x+3)⁴ go to +∞ and x = −3 is a vertical asymptote. Reveals on click.

Odd power

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:

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

The odd-power case: 1/(x−2)³ splits to −∞ and +∞, the two-sided limit DNE, yet x = 2 is still a vertical asymptote. Reveals on click.

$x = -4$ (smooth): $\displaystyle\lim_{x \to -4} f = 0$ , and $f(-4) = 0$ .

$x = -2$ (hole): $\displaystyle\lim_{x \to -2} f = 3$ , 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 exhibiting a smooth point, a hole, a jump, and a vertical asymptote

Figure 10: One graph of $f$ showing all four behaviors at once.

A single graph carrying all four behaviors — smooth, hole, jump, asymptote — read off one point at a time on click.

2.2

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

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 it agrees, the limit exists; when it splits or blows up, we don't stop at "DNE" — we name how.

Figure 10: The four verdicts, read off one graph.

The closing ledger: four points, four verdicts, over a ghost 2.2. The limit reads the neighborhood, never the point — and we name how it fails. Abstract reveals on click.

The Limit of a Function

Learning objectives

Closing in on $x = 2$

The limit, intuitively

Why the hole?

Example 2.2.1 — a limit from a table

Try It Now 2.2.1

Example 2.2.3 — a limit from a graph

Theorem 2.2.1 — two important limits

Two ways a limit can fail

Oscillation

Blow-up

Example 2.2.4 — a limit that fails

When "DNE" hides the story

Definition 2.2.2 — one-sided limits

Example 2.2.5 — one-sided limits

Try It Now 2.2.4

Theorem 2.2.2 — sides & two-sided

Back to the blow-up

Infinite limits & vertical asymptotes

Example 2.2.6 — an infinite limit

Theorem 2.2.3 — infinite limits, $1/(x-a)^n$

Example 2.2.7 — a vertical asymptote

Try It Now 2.2.6

Example 2.2.8 — four behaviors

The question, answered

The Limit of a Function

Learning objectives

Closing in on x=2x = 2x=2

The limit, intuitively

Why the hole?

Example 2.2.1 — a limit from a table

Try It Now 2.2.1

Example 2.2.3 — a limit from a graph

Theorem 2.2.1 — two important limits

Two ways a limit can fail

Oscillation

Blow-up

Example 2.2.4 — a limit that fails

When "DNE" hides the story

Definition 2.2.2 — one-sided limits

Example 2.2.5 — one-sided limits

Try It Now 2.2.4

Theorem 2.2.2 — sides & two-sided

Back to the blow-up

Infinite limits & vertical asymptotes

Example 2.2.6 — an infinite limit

Theorem 2.2.3 — infinite limits, 1/(x−a)n1/(x-a)^n1/(x−a)n

Example 2.2.7 — a vertical asymptote

Try It Now 2.2.6

Example 2.2.8 — four behaviors

The question, answered

Closing in on $x = 2$

Theorem 2.2.3 — infinite limits, $1/(x-a)^n$