ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every polynomial has $\lim_{x \to c} p(x) = p(c)$ , and rational functions do so away from the zeros of the denominator

Example

For a list $a_0, \dots, a_{n-1}$ of reals write

$p(x) \;:=\; \sum_{j < n} a_j x^j$

for the finite sum of Finite sums and finite products, by recursion applied to the list $j \mapsto a_j x^j$ , with powers as in Integer powers $a^m$ . So $\sum_{j < 0}$ is the empty sum $0$ , and $p$ is a function $\mathbb{R} \to \mathbb{R}$ ; these are the polynomial functions.

Claim 1. For every polynomial function $p$ and every $c \in \mathbb{R}$ , the limit of $p$ at $c$ exists and

$\lim_{x \to c} p(x) \;=\; p(c) .$

Claim 2. Let $p$ and $q$ be polynomial functions and let $c \in \mathbb{R}$ satisfy $q(c) \ne 0$ . Put $A_q := \{\, x \in \mathbb{R} : q(x) \ne 0 \,\}$ . Then $c \in A_q$ , the point $c$ is a limit point of $A_q$ , the quotient $p/q$ is defined on $A_q$ , its limit at $c$ exists, and

$\lim_{x \to c} (p/q)|_{A_q}(x) \;=\; \frac{p(c)}{q(c)} .$

Everything is read off from Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero once two trivial limits are in hand: that of a constant function and that of the identity. Note that claim 1 is exactly the statement that $\lim_{x \to c} p(x) = p(c)$ , the equality that FALSE: $\lim_{x \to c} f(x) = f(c)$ whenever both sides exist shows is not automatic; for polynomials it is a theorem, and the algebra of limits is what proves it.

Facts & Assumptions

Given: A list $a_0, \dots, a_{n-1}$ of reals and the polynomial function $p(x) = \sum_{j<n} a_j x^j$ ; a second polynomial function $q$ ; and a real $c$ (Finite sums and finite products, by recursion, Integer powers $a^m$ ).

[L1]

The limit condition (The $\varepsilon$ - $\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ ): $\lim_{x \to c} h(x) = P$ means that for every real $\varepsilon > 0$ there is a real $\delta > 0$ such that every $x$ in the domain of $h$ with $0 < |x - c| < \delta$ satisfies $|h(x) - P| < \varepsilon$ .

[L2]

Algebra of function limits: at a limit point $c$ of the common domain, the limits of $f + g$ , of $\alpha f$ and of $fg$ exist and equal $L + M$ , $\alpha L$ and $LM$ ; and if $M \ne 0$ the limit of $(f/g)$ restricted to $\{g \ne 0\}$ exists and equals $L/M$ (Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero).

[L3]

Every real is a limit point of $\mathbb{R}$ , punctured neighbourhoods being never empty (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$ , The $\varepsilon$ -neighbourhood and the punctured $\varepsilon$ -neighbourhood of a point of $\mathbb{R}$ ).

[L4]

Finite sums: $\sum_{j<0} a_j = 0$ and $\sum_{j<n+1} a_j = \bigl(\sum_{j<n} a_j\bigr) + a_n$ (Finite sums and finite products, by recursion).

[L5]

Powers: $x^0 = 1$ and $x^{j+1} = x^j \cdot x$ for every $x \in \mathbb{R}$ and $j \in \mathbb{N}$ (Integer powers $a^m$ ).

[L6]

Induction principle on $\mathbb{N}$ (The principle of mathematical induction).

[L7]

Sign preservation: if the limit of $g$ at $c$ is nonzero then $c$ is a limit point of $\{\, x : g(x) \ne 0 \,\}$ (If $\lim_{x \to c} f(x) = L \ne 0$ then $|f| > |L|/2$ on a punctured neighbourhood of $c$ ; in particular if $L > 0$ then $f > L/2 > 0$ there).

[L8]

Absolute value: $|0| = 0$ ; and field arithmetic (Basic properties of the absolute value, Field).

Verification

technique · direct

1.1

Every $c \in \mathbb{R}$ is a limit point of $\mathbb{R}$ , so [L1] and [L2] apply at $c$ to functions defined on $\mathbb{R}$ .

1.2

A constant function $x \mapsto a$ has limit $a$ at $c$ : $|a - a| = |0| = 0 < \varepsilon$ for every real $\varepsilon > 0$ , any $\delta$ serving.

L1L8

1.3

The identity function $x \mapsto x$ has limit $c$ at $c$ : given a real $\varepsilon > 0$ , take $\delta := \varepsilon$ ; then $0 < |x - c| < \delta$ gives $|x - c| < \varepsilon$ .

2.1

For every $j \in \mathbb{N}$ the function $x \mapsto x^j$ has limit $c^j$ at $c$ . This is an induction on $j$ [L6]. For $j = 0$ the function is the constant $1$ by [L5], and step 1.2 applies with $a = 1 = c^0$ . If the claim holds for $j$ , then $x^{j+1} = x^j \cdot x$ by [L5], and the product rule of [L2] applied to $x \mapsto x^j$ and the identity gives limit $c^j \cdot c = c^{j+1}$ .

step 1.2step 1.3L2L5L6

3.1

For every $j \in \mathbb{N}$ the function $x \mapsto a_j x^j$ has limit $a_j c^j$ at $c$ , by the scalar rule of [L2] applied to step 2.1 with $\alpha = a_j$ .

step 2.1L2

4.1

For every $n \in \mathbb{N}$ the function $x \mapsto \sum_{j<n} a_j x^j$ has limit $\sum_{j<n} a_j c^j$ at $c$ . This is an induction on $n$ [L6]. For $n = 0$ both the function and the asserted limit are the empty sum $0$ by [L4], and step 1.2 applies. If the claim holds for $n$ , then $\sum_{j<n+1} a_j x^j = \bigl(\sum_{j<n} a_j x^j\bigr) + a_n x^n$ by [L4], and the sum rule of [L2] applied to the inductive hypothesis and step 3.1 gives limit $\bigl(\sum_{j<n} a_j c^j\bigr) + a_n c^n = \sum_{j<n+1} a_j c^j$ . Taking the given $n$ , the limit of $p$ at $c$ exists and equals $p(c)$ : claim 1.

step 1.2step 3.1L2L4L6

5.1

Now let $q$ be a polynomial function with $q(c) \ne 0$ and put $A_q := \{\, x \in \mathbb{R} : q(x) \ne 0 \,\}$ . By step 4.1 the limit of $q$ at $c$ exists and equals $q(c) \ne 0$ , so [L7] gives that $c$ is a limit point of $A_q$ ; and $c \in A_q$ because $q(c) \ne 0$ .

step 4.1L7

6.1

The quotient rule of [L2], applied on $\mathbb{R}$ to $p$ and $q$ with $\lim_{x \to c} q(x) = q(c) \ne 0$ , gives that the limit of $(p/q)|_{A_q}$ at $c$ exists and equals $p(c)/q(c)$ : claim 2.

step 1.1step 4.1step 5.1L2∎

Remarks

Two inductions, and why they are separate. The first builds the monomials from the identity by repeated multiplication; the second builds the polynomial from the monomials by repeated addition. Each is an induction on the recursion clause of the object it builds (Integer powers $a^m$ and Finite sums and finite products, by recursion respectively), and neither can be replaced by dots.
Index hygiene. The sum is written $\sum_{j<n}$ , whose first index is $j = 0$ and whose empty case $n = 0$ is the zero function; the base case of step 4.1 is that empty case, and $x^0 = 1$ holds for every real $x$ including $0$ (Integer powers $a^m$ ), so no index or value is left undefined.
What claim 2 does not say. It says nothing at a zero of $q$ . There the quotient is undefined, and whether it has a limit depends on $p$ as well. It may have one: on $\{\, x : x \ne 0 \,\}$ the quotient $x^2/x$ equals $x$ , whose limit at $0$ is $0$ by step 1.3 and The limit at $c$ depends only on the restriction of $f$ to a punctured neighbourhood of $c$ , and passes to any subset of the domain having $c$ as a limit point. It may also fail to have one. Nothing on this page decides such cases in general.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 77 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis I, §3.1: Limits of functions (standard reference, not scraped)
Polynomial (Wikipedia) (standard reference, not scraped)
Limit of a function (Wikipedia) (standard reference, not scraped)
J. Lebl, Basic Analysis I, §3.1 (standard reference, not scraped)

Every polynomial has lim⁡x→cp(x)=p(c)\lim_{x \to c} p(x) = p(c)limx→c​p(x)=p(c), and rational functions do so away from the zeros of the denominator