TheoremStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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.

The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$

Statement

$\exp (x) \to + \infty (x \to + \infty), \exp (x) \to 0 (x \to - \infty),$ and the range of $\exp$ is contained in $(0, \infty)$ and is unbounded above with infimum $0$ .

Facts & Assumptions

Given: The exponential series.

[L1]

For $x \geq 0$ , every exponential-series term is nonnegative, so its sum dominates every partial sum and in particular $\exp (x) \geq 1 + x$ (The real exponential function and the number $e$ by a power series, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum).

[L2]

$\exp (- x) = 1 / \exp (x) > 0$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ).

[L3]

Finite and infinite limits of functions at infinity have the quantified definitions in Limits at $+ \infty$ and $- \infty$ , and infinite limits at a point.

Proof

technique · direct

1.1L1L3

Given a real $M$ , every $x > \max {0, M - 1}$ satisfies $\exp (x) \geq 1 + x > M$ . Hence $\exp (x) \to + \infty$ .

1.2L1L2choose

Given $ε > 0$ , choose $X > 0$ with $1 + X > 1 / ε$ . If $x < - X$ , then $- x > X$ , so [L1] gives $\exp (- x) \geq 1 - x > 1 + X > 1 / ε$ ; [L2] yields $0 < \exp (x) < ε$ .

The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The exponential tends to +∞+∞ at +∞+∞ and to 00 at −∞−∞

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$