Limits and Continuity

⚡ Key Takeaways

A limit describes the value a function approaches as its input approaches some point; continuity means the function's value there equals that limit, with no jumps, holes, or breaks. Limits formalize "getting arbitrarily close" and underpin all of calculus. Used wherever instantaneous rates, areas, or infinite processes appear — physics, engineering, economics. Developed rigorously by Cauchy and Weierstrass in the 19th century.

💡 The Core Insight

A limit cares only about the behavior near a point, never the value at it. We deliberately exclude $x = a$ from consideration. This separation is what lets us divide by zero "in disguise" — as in $\frac{x^2-1}{x-1}$ at $x=1$ — and still extract a meaningful answer. Continuity then bolts the value back on: it is the special, well-behaved case where nearby behavior and actual value agree.

DefinitionLimit of a Function (intuitive)

We write

1

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

Derivation hint: this is notation, not a derived result — read it as "as $x$ tends to $a$ , $f(x)$ tends to $L$ ." if $f(x)$ approaches the single finite number $L$ as $x$ approaches $a$ from both sides, without $x$ ever equaling $a$ . Symbols: $f$ is the function, $x$ the independent variable, $a$ the point being approached (need not be in the domain), $L$ the limiting value.

DefinitionThe ε–δ Definition (rigorous)

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

2

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

Derivation hint: $\varepsilon$ is the target closeness on the output; $\delta$ is the required closeness on the input; the strict $0<|x-a|$ excludes the point itself. Here $\varepsilon$ ("how close to $L$ we demand") is chosen first by a skeptic; we must respond with a $\delta$ ("how close to $a$ we need $x$ ") that guarantees it.

DefinitionOne-Sided Limits

The left-hand limit and right-hand limit:

3

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

Derivation hint: restrict the ε–δ condition to $a-\delta < x < a$ (left) or $a < x < a+\delta$ (right). The two-sided limit exists iff both one-sided limits exist and are equal: $L_- = L_+ = L$ .

DefinitionContinuity at a Point

A function $f$ is continuous at $a$ if all three conditions hold:

4

\quad f(a)\text{ is defined}, \quad \lim_{x\to a} f(x)\text{ exists}, \quad \lim_{x\to a} f(x) = f(a)

Derivation hint: combine the existence of value [1st], existence of limit [Eq. 1], and their equality into one statement. $f$ is continuous on an interval if it is continuous at every point of that interval.

TheoremLimit Laws

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

5

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

Derivation hint: bound $|(f+g)-(L+M)| \le |f-L|+|g-M|$ via the triangle inequality, then make each piece $<\varepsilon/2$ .

6

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

Derivation hint: add and subtract $L\,g(x)$ , then bound $|fg - LM| \le |g||f-L| + |L||g-M|$ .

7

\quad \lim_{x\to a}\frac{f(x)}{g(x)} = \frac{L}{M}, \quad M \ne 0

Derivation hint: from $M\ne0$ , $g$ stays bounded away from 0 near $a$ ; control $\left|\frac{f}{g}-\frac{L}{M}\right|$ using that bound.

TheoremSqueeze (Sandwich) Theorem

If $g(x) \le f(x) \le h(x)$ for all $x$ near $a$ (except possibly at $a$ ), and

8

\quad \lim_{x\to a} g(x) = \lim_{x\to a} h(x) = L \;\implies\; \lim_{x\to a} f(x) = L

Derivation hint: given $\varepsilon$ , both $g$ and $h$ lie within $\varepsilon$ of $L$ ; the trapped $f$ has no choice but to follow.

ProofThe Standard Limit

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

Steps using the Squeeze Theorem and unit-circle geometry: 1. For $0 < x < \tfrac{\pi}{2}$ , compare areas in a unit circle: the triangle with area $\tfrac12\sin x$ , the sector with area $\tfrac12 x$ , and the larger triangle with area $\tfrac12\tan x$ . 2. This gives $\sin x < x < \tan x$ . 3. Divide through by $\sin x > 0$ : $\;1 < \dfrac{x}{\sin x} < \dfrac{1}{\cos x}$ . 4. Take reciprocals (reversing inequalities): $\;\cos x < \dfrac{\sin x}{x} < 1$ . 5. Since $\frac{\sin x}{x}$ and $\cos x$ are even, the same bound holds for $-\tfrac{\pi}{2} < x < 0$ . 6. As $x\to 0$ , $\cos x \to 1$ and the constant $1\to 1$ . By the Squeeze Theorem (Eq. 8), the trapped quantity satisfies

9

\quad \lim_{x\to 0}\frac{\sin x}{x} = 1

Derivation hint: the geometric area inequality from step 2 is the entire engine; squeezing finishes it.

∎

ProofA Removable Discontinuity Has a Limit

Consider $f(x) = \dfrac{x^2 - 1}{x - 1}$ , undefined at $x = 1$ . 1. For $x \ne 1$ , factor the numerator: $x^2 - 1 = (x-1)(x+1)$ . 2. Cancel the common factor (legal because $x \ne 1$ , so $x - 1 \ne 0$ ): $f(x) = x + 1$ . 3. The function $x + 1$ is a polynomial, continuous everywhere, so $\lim_{x\to 1}(x+1) = 2$ . 4. Therefore $\lim_{x\to 1} f(x) = 2$ , even though $f(1)$ does not exist.

10

\quad \lim_{x\to 1}\frac{x^2-1}{x-1} = 2

Derivation hint: the limit ignores $x=1$ , so cancellation is valid throughout the neighborhood used. This is a removable discontinuity: redefining $f(1) = 2$ makes $f$ continuous.

∎

mathematician · Augustin-Louis Cauchy | 1789–1857

Cauchy transformed calculus from a collection of useful but shaky procedures into a rigorous discipline. In his Cours d'analyse (1821), he based the entire subject on the limit concept, defining continuity, derivatives, and integrals through limits rather than vague "infinitesimals." His phrasing — "the values successively attributed to a variable approach indefinitely a fixed value" — is the verbal ancestor of the ε–δ definition that Karl Weierstrass later made fully precise. Cauchy's insistence on convergence and careful inequalities set the standard for all modern analysis.

🧠 Intuition

Think of a limit as the answer to: "Where is the function clearly headed?" — not "Where does it land?" Imagine walking along the graph toward $x = a$ . If both your left-foot path and right-foot path aim at the same height $L$ , the limit is $L$ , regardless of whether there's a pothole (hole), a cliff (jump), or a flagpole (a point plotted somewhere else) exactly at $a$ . Continuity is the absence of all three pathologies: no pothole, no cliff, no misplaced flagpole. The pen-without-lifting picture is exactly right at this level — a continuous function on an interval can be traced in one unbroken stroke. The ε–δ game makes "headed toward" exact. Your opponent draws a horizontal band of width $2\varepsilon$ around $L$ and dares you. You reply by drawing a vertical strip of width $2\delta$ around $a$ such that the graph inside your strip (minus the center line) stays inside their band. If you can always win, no matter how thin their band, the limit is $L$ .

Figure

A removable hole, a jump, and a continuous point — three behaviors at x=a

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ExampleEvaluating a Limit by Factoring

Given: Find $\displaystyle \lim_{x\to 3}\frac{x^2 - 9}{x^2 - x - 6}$ .

Step 1. Substitute $x = 3$ directly: numerator $= 0$ , denominator $= 9 - 3 - 6 = 0$ . Indeterminate form $\tfrac{0}{0}$ — factoring needed.

Step 2. Factor both:

Numerator: $x^2 - 9 = (x-3)(x+3)$ .
Denominator: $x^2 - x - 6 = (x-3)(x+2)$ .

Step 3. Cancel the common factor $(x-3)$ , valid since $x\ne 3$ in the limit:

11

\quad \frac{(x-3)(x+3)}{(x-3)(x+2)} = \frac{x+3}{x+2}

Derivation hint: cancellation is legitimate because $x$ approaches but never reaches 3.

Step 4. Now substitute $x = 3$ into the simplified expression: $\dfrac{3+3}{3+2} = \dfrac{6}{5}$ .

\boxed{\lim_{x\to 3}\frac{x^2-9}{x^2-x-6} = \frac{6}{5} = 1.2}

ExampleContinuity of a Piecewise Function

Given: Determine the value of $k$ making $f$ continuous at $x = 2$ , where

f(x) = \begin{cases} 3x + 1 & x < 2 \\ kx^2 - 1 & x \ge 2 \end{cases}

Step 1. Left-hand limit: $\displaystyle\lim_{x\to 2^-} f(x) = 3(2) + 1 = 7$ .

Step 2. Right-hand limit (and the value $f(2)$ ): $\displaystyle\lim_{x\to 2^+} f(x) = k(2)^2 - 1 = 4k - 1$ .

Step 3. For continuity (Eq. 4), the two one-sided limits must agree, and equal $f(2)$ :

12

\quad 7 = 4k - 1

Derivation hint: set left-hand limit = right-hand limit, the continuity-at-a-point requirement from Eq. 4.

Step 4. Solve: $4k = 8 \implies k = 2$ .

Check: With $k = 2$ , $f(2) = 2(4) - 1 = 7 =$ left limit. ✓

\boxed{k = 2}

⚠️ Common mistake

"The limit equals the function value." False in general. The limit ignores $f(a)$ entirely; equality holds only when $f$ is continuous at $a$ . You may substitute directly only after confirming continuity (true for all polynomials, and for rational/trig/exponential functions wherever they are defined).

" $\frac{0}{0}$ means the limit is 0 (or undefined, or 1)." False. $\frac{0}{0}$ is an indeterminate form — it signals that more work (factoring, rationalizing, the Squeeze Theorem) is needed; the true limit can be any number, $\infty$ , or nonexistent.

"If the limit doesn't exist, the function is undefined there." Confusion of two ideas. $f(a)$ can be perfectly defined while the limit fails to exist (a jump), and the limit can exist while $f(a)$ is undefined (a hole).

Counterexample

Consider $f(x) = \sin\!\left(\dfrac{1}{x}\right)$ as $x \to 0$ . As $x$ shrinks, $\tfrac1x$ races to $+\infty$ , so $f$ oscillates between $-1$ and $+1$ infinitely many times in any neighborhood of $0$ . No single value $L$ is approached — the limit does not exist, even though the function stays bounded. This kills the naïve belief that "bounded near $a$ " implies "has a limit at $a$ ." Contrast: $g(x) = x\sin\!\left(\dfrac1x\right)$ does have limit $0$ at $x=0$ , because $-|x| \le g(x) \le |x|$ and the Squeeze Theorem (Eq. 8) forces $g \to 0$ . Multiplying by the shrinking factor $x$ tames the oscillation.

connection

Limits are the foundation on which the rest of calculus stands. The Derivative is defined as a limit of difference quotients, $f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ — and a function must be continuous at $a$ to be differentiable there (though continuity alone is not enough). The Definite Integral is a limit of Riemann Sums as the partition width shrinks to zero. The Intermediate Value Theorem and Extreme Value Theorem both require continuity as their hypothesis, showing why continuity is the gatekeeper to calculus's deepest theorems. At the level above, Sequences and Series reuse the identical limit idea for discrete indices instead of a continuous variable.

🔬 Where this appears

⏳ History

- c. 450 BCE — Zeno of Elea poses paradoxes (Achilles and the tortoise) exposing the puzzle of infinite subdivision, the conceptual seed of limits.

- c. 250 BCE — Archimedes uses the method of exhaustion to find areas and the value of

\pi

, effectively taking limits without the vocabulary.

- 1660s–1680s — Newton ("ultimate ratios") and Leibniz (infinitesimals

dx

) independently invent calculus, powerfully but without rigorous foundations.

- 1734 — Bishop Berkeley's The Analyst mocks infinitesimals as "ghosts of departed quantities," demanding rigor.

- 1817 — Bernard Bolzano gives an early rigorous definition of continuity and proves a version of the Intermediate Value Theorem.

- 1821 — Augustin-Louis Cauchy founds calculus on limits in Cours d'analyse, defining continuity and convergence carefully.

- 1854–1872 — Karl Weierstrass perfects the static ε–δ definition, removing all reference to motion or infinitesimals, and constructs a continuous-but-nowhere-differentiable function.

- 1872 — Richard Dedekind and Georg Cantor rigorously construct the real numbers, supplying the completeness limits silently rely on.

- 1960s — Abraham Robinson's non-standard analysis rehabilitates infinitesimals rigorously, vindicating Leibniz's intuition three centuries later.

open-problem

The everyday "draw without lifting your pen" picture, while perfect at this level, conceals genuine subtleties. Weierstrass's function is continuous everywhere yet differentiable nowhere — an unbroken curve so jagged it has no tangent at any point, impossible to truly "draw." This shows continuity is far weaker and stranger than intuition suggests. Deeper still: limits depend on the completeness of the real numbers. Over the rationals alone, a sequence can crowd toward a "gap" (like $\sqrt2$ ) with no limit in the system — completeness is precisely what plugs these holes. Modern mathematics generalizes the limit far beyond the real line: in topology, limits are defined via open neighborhoods without any notion of distance, and concepts like uniform continuity, nets, and filters extend the idea to spaces where " $\varepsilon$ " and " $\delta$ " no longer make sense. The intuitive limit a student first meets is the gateway to one of mathematics' most far-reaching abstractions.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also