Limit of \( \displaystyle \frac{12 x^{2} + 1}{4 x^{2} - 4} \) as \( x \to \infty \)
Problem 1.129 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{12 x^{2} + 1}{4 x^{2} - 4} \).
- \[ \lim_{x \to \infty}\left(\frac{12 x^{2} + 1}{4 x^{2} - 4}\right) \]limitStart with the given limit.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{12 x^{2}}{4 x^{2} - 4} + \frac{1}{4 x^{2} - 4}\right) \]algebraSplit the fraction into two parts.✓ Proved
- \[ = \lim_{x \to \infty}\left(3 + \frac{1}{4 x^{2} - 4}\right) \]simplify simplifySimplify the first term by canceling x**2. The first term simplifies to 3.✓ Proved
- \[ = \lim_{x \to \infty} 3 \]limit-lawThe limit of the second term is 0 as x approaches infinity.✓ Proved
- \[ = 3 \]limitEvaluate the final limit.✓ Proved
Answer \( 3 \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy took the limit from both sides and got the stated value |
Reviewers
gpt-oss:20b: fail (error) — Step 3 incorrectly simplifies (12*x**2)/(4*x**2 - 4) to (12*x**2)/(4*x**2), dropping the -4 term. This applies two changes at once and is not a valid simplification. The correct approach is to factor out 4*x**2 from the denominator before simplifying.qwen3.6:27b-mlx: fail (error) — Step 3 is mathematically incorrect: (12*x**2)/(4*x**2 - 4) does not simplify to (12*x**2)/(4*x**2). The solver incorrectly dropped the '-4' in the denominator, which is not a valid algebraic simplification.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is mathematically incorrect: (12*x**2)/(4*x**2 - 4) does not simplify to (12*x**2)/(4*x**2). The solver incorrectly dropped the '-4' in the denominator, which is not a valid algebraic simplification.gpt-oss:20b: fail (error) 2026-10-03 — Step 3 incorrectly simplifies (12*x**2)/(4*x**2 - 4) to (12*x**2)/(4*x**2), dropping the -4 term. This applies two changes at once and is not a valid simplification. The correct approach is to factor out 4*x**2 from the denominator before simplifying.qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is mathematically incorrect: it changes the denominator of the first term from (4*x**2 - 4) to (4*x**2) without justification, effectively altering the function rather than simplifying it. This is not a valid algebraic manipulation.gpt-oss:20b: fail (error) 2026-10-03 — Step 3 incorrectly simplifies (12*x**2)/(4*x**2 - 4) to (12*x**2)/(4*x**2). This applies two rules at once (cancellation and ignoring the -4 term) and is not algebraically valid. The correct approach is to factor 4*x**2 out of the denominator before simplifying.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.