lean_workbook_plus_68866

⚠️ 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.

Explain the step where the expression is transformed from $\lim\limits_{x\rightarrow \infty} x(\frac {x} {\sqrt{x^2+2x}+\sqrt{x^2+x}} - \frac {x} {\sqrt{x^2+x} + x})$ to $\lim\limits_{x\rightarrow \infty} x^2 (\frac {-2x} {(\sqrt{x^2+2x}+\sqrt{x^2+x})(\sqrt{x^2+x} + x)(x + \sqrt{x^2+2x})}) / (\frac {x} {\sqrt{x^2+2x}+\sqrt{x^2+x}} + \frac {x} {\sqrt{x^2+x} + x})$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_68866 : ∀ x : ℝ, x ≠ 0 → x * (x / (Real.sqrt (x ^ 2 + 2 * x) + Real.sqrt (x ^ 2 + x)) - x / (Real.sqrt (x ^ 2 + x) + x)) = x ^ 2 * (-2 * x) / ((Real.sqrt (x ^ 2 + 2 * x) + Real.sqrt (x ^ 2 + x)) * (Real.sqrt (x ^ 2 + x) + x) * (x + Real.sqrt (x ^ 2 + 2 * x)))   :=  by sorry

State the rationalization identity on x>0, and express the source transformation as equality of limits at+∞. The additional divided expression has the same limit−1/4 because its extra denominator tends to1; it is not a pointwise identity.

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

Proposed corrected statement

State the rationalization identity on x>0, and express the source transformation as equality of limits at+∞. The additional divided expression has the same limit−1/4 because its extra denominator tends to1; it is not a pointwise identity.

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.