lean_workbook_plus_25848

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Using L Hospital rule,we have,\n\n $\lim_{x\\to 0}\\frac {x+1}{\\sqrt{2x+1}} + \\sqrt{2x+1} - \\frac{x^2+2009}{3x+1}^{\\frac{2}{3}} - 2x(3x+1)^{\\frac{1}{3}}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_25848 : ∀ ε : ℝ, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x : ℝ, x ∈ Set.Ioo (-δ) δ → |(x + 1) / Real.sqrt (2 * x + 1) + Real.sqrt (2 * x + 1) - (x ^ 2 + 2009) / (3 * x + 1) ^ (2 / 3) - 2 * x * (3 * x + 1) ^ (1 / 3)| < ε   :=  by sorry

An unspecified limit was encoded as convergence to0, although the expression at0 and its nearby continuous limit are−2007; fractional exponents also collapse.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Confirmed directly from the Lean text: the numeric-literal exponents carry no type ascription, so they elaborate at type ℕ where division truncates (^ (1 / 3) → ^0; ^ (2 / 3) → ^0).

Proposed corrected statement

Recover the intended parentheses and use genuine real exponents near0. For the displayed expression prove its limit is−2007, by continuity on a small interval where the radicands/bases are positive; subtract that limit in the ε-condition.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.