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

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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

1.1

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

L1L3

1.2

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 $0 < \exp (x) < ε$ .

L1L2choose

2.1

The range assertions follow from positivity and the two limit conclusions.

step 1.1step 1.2L2∎

Depends on

Used by

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)

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