Limits

⚡ Key Takeaways

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.

💡 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?"

DefinitionIntuitive (Informal) Limit

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

1

\quad \lim_{x \to a} 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

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

DefinitionThe Precise

\varepsilon

–

\delta

Definition (Cauchy–Weierstrass)

$\lim_{x \to a} f(x) = L$ means: for every $\varepsilon > 0$ there exists a $\delta > 0$ such that

2

\quad 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon.

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

Derivation hint: Translate "as close as we like" into "for every $\varepsilon$ " and "sufficiently close" into "there exists $\delta$ "; the order of quantifiers is the whole game.

DefinitionOne-Sided Limits

The right-hand and left-hand limits are

3

\quad \lim_{x \to a^+} f(x) = L_+ \quad\text{and}\quad \lim_{x \to a^-} 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)

Derivation hint: Restrict the $\varepsilon$ – $\delta$ condition to $0 < x-a < \delta$ (right) or $0 < a-x < \delta$ (left).

TheoremExistence Criterion

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

4

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

Derivation hint: A neighbourhood of $a$ splits into left and right parts; controlling both simultaneously is equivalent to controlling each.

TheoremLimit Laws

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

5

\quad \lim_{x \to a}\big[f(x) \pm g(x)\big] = L \pm M, \qquad \lim_{x \to a}\big[f(x)\,g(x)\big] = L\,M,

6

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

Derivation hint: Each follows from the $\varepsilon$ – $\delta$ definition using the triangle inequality $|f+g - (L+M)| \le |f-L| + |g-M|$ .

ProofThe Limit Laws from

\varepsilon

–

\delta

— Sum Rule

We prove $\lim_{x\to a}[f(x)+g(x)] = L+M$ . 1. Setup. Let $\varepsilon > 0$ be given. We must produce $\delta > 0$ forcing $|f(x)+g(x)-(L+M)| < \varepsilon$ . 2. Split the error. By the triangle inequality,

7

\quad |f(x)+g(x)-(L+M)| \le |f(x)-L| + |g(x)-M|.

Derivation hint: Group as $(f-L)+(g-M)$ and apply $|p+q|\le|p|+|q|$ . 3. Use the hypotheses. Since $\lim f = L$ , there is $\delta_1$ with $0<|x-a|<\delta_1 \Rightarrow |f(x)-L| < \varepsilon/2$ . Since $\lim g = M$ , there is $\delta_2$ with $0<|x-a|<\delta_2 \Rightarrow |g(x)-M| < \varepsilon/2$ . 4. Combine. Let $\delta = \min(\delta_1, \delta_2)$ . Then $0<|x-a|<\delta$ guarantees both bounds, so the right side of [7] is less than $\varepsilon/2 + \varepsilon/2 = \varepsilon$ . 5. Conclusion. We exhibited a working $\delta$ for arbitrary $\varepsilon$ , proving the sum law. $\blacksquare$

∎

ProofA Concrete

\varepsilon

–

\delta

Verification:

\lim_{x\to 3}(2x+1)=7

1. Goal. Given $\varepsilon>0$ , find $\delta$ so that $0<|x-3|<\delta \Rightarrow |(2x+1)-7|<\varepsilon$ . 2. Simplify the output gap. $|(2x+1)-7| = |2x-6| = 2|x-3|$ . 3. Choose $\delta$ . We need $2|x-3| < \varepsilon$ , i.e. $|x-3| < \varepsilon/2$ . So pick $\delta = \varepsilon/2$ . 4. Verify. If $0<|x-3|<\delta=\varepsilon/2$ , then $|(2x+1)-7| = 2|x-3| < 2\cdot(\varepsilon/2) = \varepsilon$ . $\blacksquare$

∎

mathematician · 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 $\varepsilon$ – $\delta$ method. Karl Weierstrass later (1860s) crystallized Cauchy's verbal definitions into the fully symbolic $\varepsilon$ – $\delta$ form used today, completing the "arithmetization of analysis" that banished mystical infinitesimals from the foundations of calculus.

🧠 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 $\varepsilon$ – $\delta$ game makes this a duel: a skeptic names a tiny target band of half-width $\varepsilon$ around $L$ ; you must respond with a small window of half-width $\delta$ 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 $\varepsilon$ , the limit is genuinely $L$ . The "punctured" condition $0<|x-a|$ encodes the deepest lesson: we cover our eyes at $x=a$ exactly.

Figure

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

ExampleIndeterminate Form: Factoring

Given: Evaluate $\displaystyle \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ .

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

8

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

Derivation hint: The factor $x-2$ cancels because the limit ignores $x=2$ , where the original is undefined.

Step 3 — Substitute into the simplified form. $\lim_{x\to 2}(x+2) = 2+2 = 4$ .

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

ExampleA Trigonometric Limit

Given: Evaluate $\displaystyle \lim_{x \to 0} \frac{\sin(3x)}{x}$ (with $x$ in radians).

Step 1 — Recall the standard limit. It is known that $\lim_{u\to 0}\frac{\sin u}{u} = 1$ (the squeeze theorem gives this).

Step 2 — Engineer the matching form. Multiply and divide by $3$ :

9

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

Derivation hint: We force the denominator to match the argument of sine so the standard limit applies; let $u = 3x \to 0$ .

Step 3 — Take the limit. As $x\to 0$ , $u=3x\to 0$ , so $\frac{\sin(3x)}{3x}\to 1$ . Hence the expression $\to 3\cdot 1 = 3$ .

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

⚠️ Common mistake

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.).

CounterexampleA Limit That Fails to Exist

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

10

\quad \lim_{x\to 0^+} \frac{|x|}{x} = 1 \neq -1 = \lim_{x\to 0^-} \frac{|x|}{x}.

Derivation hint: $|x|=x$ 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.

connection

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.

🔬 Where this appears

⏳ History

- 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

\varepsilon

–

\delta

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.

open-problem

The $\varepsilon$ – $\delta$ 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 $\varepsilon$ there exists $\delta$ ") 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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also