Limits

A limit describes the value that a function $f (x)$ approaches as its input $x$ approaches some point, even if the function is undefined there. Written $lim_{x \to a} f (x) = L$ , it captures behaviour near a point, not at it. Limits are the foundation of calculus — continuity, derivatives, and integrals are all defined through them. Formalized in the 19th century by Cauchy and Weierstrass.

A limit answers the question: "As $x$ gets arbitrarily close to $a$ , what single number does $f (x)$ get arbitrarily close to?" The crucial subtlety is that the limit depends only on values of $f$ near $a$ , never on $f (a)$ itself — the function need not even be defined at $a$ . Limits matter because they make rigorous the intuitive notion of "approaching," allowing us to handle quantities like instantaneous velocity ( $0/0$ in disguise) and the area under a curve. The surprising fact: a limit can exist at a point where the function is undefined, and a function can be defined at a point yet have no limit there (e.g., a jump). This separation of "value approached" from "value attained" is the central conceptual leap of the subject.

The Core Insight

A limit is about destination, not arrival. We ask where $f (x)$ is headed as $x$ travels toward $a$ — from both sides — without caring whether $f$ ever reaches that destination or what happens exactly at $a$ . If both sides agree on a single finite target $L$ , the limit is $L$ . The entire edifice of calculus rests on replacing "what is $f$ at this point?" with "what does $f$ approach near this point?"

Formal Statement

Intuitive (Informal) Limit

We say the limit of $f (x)$ as $x$ approaches $a$ equals $L$ , written

x \to a lim f (x) = L,

if we can make $f (x)$ as close to $L$ as we like by taking $x$ sufficiently close to $a$ (but not equal to $a$ ).

$x$ — the input variable
$a$ — the point being approached (need not be in the domain of $f$ )
$f (x)$ — the output of the function
$L$ — the (finite) limit value

⟳This is a definition, not a derived result; it encodes "closeness of output controlled by closeness of input."

The Precise $\varepsilon$–$\delta$ Definition (Cauchy–Weierstrass)

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

means: for every

ε > 0

there exists a

δ > 0

such that

0 < ∣ x - a ∣ < δ ⟹ ∣ f (x) - L ∣ < ε .

$ε$ (epsilon) — a challenge: how close to $L$ we demand the output be
$δ$ (delta) — a response: how close to $a$ the input must be
$0 < ∣ x - a ∣$ — excludes $x = a$ itself (value at $a$ irrelevant)
$∣ f (x) - L ∣ < ε$ — output lands within the tolerance band

⟳Translate "as close as we like" into "for every

ε

" and "sufficiently close" into "there exists

δ

"; the order of quantifiers is the whole game.

One-Sided Limits

The right-hand and left-hand limits are

x \to a^{+} lim f (x) = L_{+} and x \to a^{-} lim f (x) = L_{-},

approaching only through $x > a$ or only through $x < a$ respectively.

$a^{+}$ — approach from above (larger values)
$a^{-}$ — approach from below (smaller values)

⟳Restrict the

ε

–

δ

condition to

0 < x - a < δ

(right) or

0 < a - x < δ

(left).

Existence Criterion

The two-sided limit exists if and only if both one-sided limits exist and are equal:

x \to a lim f (x) = L ⟺ x \to a^{-} lim f (x) = L = x \to a^{+} lim f (x) .

⟳A neighbourhood of

a

splits into left and right parts; controlling both simultaneously is equivalent to controlling each.

Limit Laws

If $lim_{x \to a} f (x) = L$ and $lim_{x \to a} g (x) = M$ (both finite), then:

x \to a lim [f (x) \pm g (x)] = L \pm M, x \to a lim [f (x) g (x)] = L M,

x \to a lim \frac{f ( x )}{g ( x )} = \frac{L}{M} (M \neq = 0), x \to a lim [c f (x)] = c L .

⟳Each follows from the

ε

–

δ

definition using the triangle inequality

∣ f + g - (L + M) ∣ \leq ∣ f - L ∣ + ∣ g - M ∣

Why It Holds / Derivation

The Limit Laws from $\varepsilon$–$\delta$ — Sum Rule

We prove $lim_{x \to a} [f (x) + g (x)] = L + M$ .

Setup. Let $ε > 0$ be given. We must produce $δ > 0$ forcing $∣ f (x) + g (x) - (L + M) ∣ < ε$ .

Split the error. By the triangle inequality,

∣ f (x) + g (x) - (L + M) ∣ \leq ∣ f (x) - L ∣ + ∣ g (x) - M ∣.

⟳Group as

(f - L) + (g - M)

and apply

∣ p + q ∣ \leq ∣ p ∣ + ∣ q ∣

Use the hypotheses. Since $lim f = L$ , there is $δ_{1}$ with $0 < ∣ x - a ∣ < δ_{1} \Rightarrow ∣ f (x) - L ∣ < ε /2$ . Since $lim g = M$ , there is $δ_{2}$ with $0 < ∣ x - a ∣ < δ_{2} \Rightarrow ∣ g (x) - M ∣ < ε /2$ .

Combine. Let $δ = min (δ_{1}, δ_{2})$ . Then $0 < ∣ x - a ∣ < δ$ guarantees both bounds, so the right side of [7] is less than $ε /2 + ε /2 = ε$ .

Conclusion. We exhibited a working $δ$ for arbitrary $ε$ , proving the sum law. $■$

A Concrete $\varepsilon$–$\delta$ Verification: $\lim_{x\to 3}(2x+1)=7$

Goal. Given $ε > 0$ , find $δ$ so that $0 < ∣ x - 3∣ < δ \Rightarrow ∣ (2 x + 1) - 7∣ < ε$ .

Simplify the output gap. $∣ (2 x + 1) - 7∣ = ∣2 x - 6∣ = 2∣ x - 3∣$ .

Choose $δ$ . We need $2∣ x - 3∣ < ε$ , i.e. $∣ x - 3∣ < ε /2$ . So pick $δ = ε /2$ .

Verify. If $0 < ∣ x - 3∣ < δ = ε /2$ , then $∣ (2 x + 1) - 7∣ = 2∣ x - 3∣ < 2 \cdot (ε /2) = ε$ . $■$

Augustin-Louis Cauchy | 1789–1857

Cauchy was the French mathematician who, in his 1821 Cours d'analyse, first gave calculus a rigorous footing based on limits rather than vague "infinitesimals." He defined continuity and convergence in terms of approaching values and pioneered the careful inequality arguments that became the $ε$ – $δ$ method. Karl Weierstrass later (1860s) crystallized Cauchy's verbal definitions into the fully symbolic $ε$ – $δ$ form used today, completing the "arithmetization of analysis" that banished mystical infinitesimals from the foundations of calculus.

Building Intuition

Picture standing at the input $x = a$ and watching the output. Now imagine walking toward $a$ along the $x$ -axis — first from the right, then from the left — and tracking where the dot $f (x)$ on the $y$ -axis is heading. The limit is the single $y$ -value the dot homes in on. The $ε$ – $δ$ game makes this a duel: a skeptic names a tiny target band of half-width $ε$ around $L$ ; you must respond with a small window of half-width $δ$ around $a$ such that every input in your window (except $a$ itself) lands inside the band. If you can always win, no matter how mean the $ε$ , the limit is genuinely $L$ . The "punctured" condition $0 < ∣ x - a ∣$ encodes the deepest lesson: we cover our eyes at $x = a$ exactly.

A function with a removable discontinuity: the limit exists even though f(a) is undefined

Worked Examples

Indeterminate Form: Factoring

Given: Evaluate

x \to 2 lim \frac{x ^{2} - 4}{x - 2}

. Step 1 — Direct substitution. Plugging

x = 2

gives

\frac{0}{0}

, an indeterminate form — no information yet. Step 2 — Factor the numerator.

x^{2} - 4 = (x - 2) (x + 2)

, so

\frac{x ^{2} - 4}{x - 2} = \frac{( x - 2 ) ( x + 2 )}{x - 2} = x + 2 (x \neq = 2) .

⟳The factor

x - 2

cancels because the limit ignores

x = 2

, where the original is undefined.

Step 3 — Substitute into the simplified form.

lim_{x \to 2} (x + 2) = 2 + 2 = 4

x \to 2 lim \frac{x ^{2} - 4}{x - 2} = 4

A Trigonometric Limit

Given: Evaluate

x \to 0 lim \frac{sin ( 3 x )}{x}

(with

x

in radians). Step 1 — Recall the standard limit. It is known that

lim_{u \to 0} \frac{s i n u}{u} = 1

(the squeeze theorem gives this). Step 2 — Engineer the matching form. Multiply and divide by

3

\frac{sin ( 3 x )}{x} = 3 \cdot \frac{sin ( 3 x )}{3 x} .

⟳We force the denominator to match the argument of sine so the standard limit applies; let

u = 3 x \to 0

Step 3 — Take the limit. As

x \to 0

u = 3 x \to 0

, so

\frac{s i n ( 3 x )}{3 x} \to 1

. Hence the expression

\to 3 \cdot 1 = 3

x \to 0 lim \frac{sin ( 3 x )}{x} = 3

Common Misconceptions

The limit is NOT the function value.

lim_{x \to a} f (x)

may differ from

f (a)

, or exist when

f (a)

does not. Substituting

x = a

only works when

f

happens to be continuous there. Treating substitution as the definition of a limit causes errors precisely at the interesting points (holes, jumps,

0/0

forms). Also: an answer of

\frac{0}{0}

is indeterminate, not "equal to 1" and not "undefined limit" — it means more work is needed (factor, rationalize, use L'Hôpital, etc.).

A Limit That Fails to Exist

Consider $f (x) = \frac{∣ x ∣}{x}$ near $x = 0$ . For $x > 0$ , $f (x) = 1$ ; for $x < 0$ , $f (x) = - 1$ . Then

x \to 0^{+} lim \frac{∣ x ∣}{x} = 1 \neq = - 1 = x \to 0^{-} lim \frac{∣ x ∣}{x} .

⟳

∣ x ∣ = x

on the right,

∣ x ∣ = - x

on the left; the one-sided limits disagree, so by [4] the two-sided limit does not exist.

A subtler case is $lim_{x \to 0} sin (1/ x)$ : it does not exist because $sin (1/ x)$ oscillates infinitely fast between $- 1$ and $1$ near $0$ , never settling toward any single value — even though both sides "fail" the same way.

Connections

Limits are the gateway to nearly all of calculus. Continuity is defined by the limit condition $lim_{x \to a} f (x) = f (a)$ . The Derivative is itself a limit — the limit of a difference quotient $\frac{f ( x + h ) - f ( x )}{h}$ as $h \to 0$ , which is generically a $0/0$ form resolved by the techniques above. The Definite Integral arises as the limit of Riemann Sums as the partition width shrinks to zero. The behaviour of Sequences and Series uses the closely related notion of the limit of a sequence, and Limits at Infinity extend the idea to describe asymptotic behaviour and horizontal asymptotes.

Real-World Applications

Instantaneous velocity — physics computes speed at an instant as the limit of average velocity over shrinking time intervals (the derivative).
Marginal cost/revenue — economics defines "marginal" quantities as limits of average rates of change, driving optimization of production.
Continuously compounded interest — the constant $e$ emerges as $lim_{n \to \infty} (1 + 1/ n)^{n}$ , underlying finance and growth models.
Numerical methods — engineering algorithms (Newton's method, finite-difference approximations) rely on limits to guarantee convergence to a true answer.
Signal processing — the response of circuits and filters at limiting frequencies (e.g., as frequency $\to 0$ or $\to \infty$ ) is found via limits.
Population & epidemic models — carrying capacities and steady states are long-term limits $lim_{t \to \infty}$ of differential-equation solutions.
Computer graphics — smooth curve rendering and level-of-detail use limiting (refined) subdivisions of polygons.

Historical Development

c. 450 BCE — Zeno of Elea poses paradoxes (Achilles and the tortoise) that implicitly demand a theory of infinite sums and limits.
c. 250 BCE — Archimedes uses the method of exhaustion to find areas and volumes, approaching answers through inscribed/circumscribed figures — a proto-limit.
1660s–1680s — Newton ("ultimate ratios") and Leibniz develop calculus using infinitesimals, powerful but logically unrigorous.
1734 — Bishop Berkeley's The Analyst attacks infinitesimals as "ghosts of departed quantities," exposing foundational gaps.
1821 — Cauchy's Cours d'analyse bases analysis on limits, defining continuity and convergence rigorously in words.
1854–1872 — Weierstrass introduces the fully formal $ε$ – $δ$ definition, eliminating reliance on geometric intuition.
1872 — Dedekind and Cantor construct the real numbers rigorously, supplying the completeness that makes limits behave well.
1960 — Abraham Robinson develops non-standard analysis, rehabilitating infinitesimals on a rigorous logical basis — an alternative route to the same limits.

Subtleties & Limits

The $ε$ – $δ$ framework is complete and rigorous for ordinary real functions, so there is no "open problem" in the research sense at this level. The genuine subtleties are conceptual and pedagogical: (1) the order of quantifiers ("for all $ε$ there exists $δ$ ") is famously hard to internalize and is where most student errors and even historical confusions lived for two centuries. (2) Limits depend on the underlying notion of "closeness" — generalizing limits to higher dimensions, abstract metric spaces, and topological spaces (via open sets and nets/filters) shows that "limit" is really a property of topology, not of formulas. (3) The existence of limits that no algorithm can compute, and pathological functions (continuous nowhere-differentiable, like Weierstrass's) discovered because the rigorous definition allowed them, revealed that intuition about "smooth approach" can badly mislead.