Limit of \( \displaystyle \frac{4 x + 1}{4 x^{2} + 3} \) as \( x \to \infty \)
Problem 1.70 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{4 x + 1}{4 x^{2} + 3} \).
- \[ \lim_{x \to \infty}\left(\frac{4 x + 1}{4 x^{2} + 3}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{\frac{4}{x} + \frac{1}{x^{2}}}{4 + \frac{3}{x^{2}}}\right) \]algebra simplifyDivide the numerator and denominator by the highest power of x in the denominator, which is x**2. Simplify the terms inside the limit.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{4}{x}\right) \left(\lim_{x \to \infty}\left(4 + \frac{3}{x^{2}}\right)\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = 0 \]limit simplifyEvaluate the limits of the individual components as x approaches infinity. Final evaluation of the fraction.✓ Proved
Answer \( 0 \)
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 + 3 = 0 undefined where 4 + 3/x**2 = 0 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4 + 3/x**2 = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4 + 3/x**2 = 0 undefined where x = 0 undefined where Limit(4 + 3/x**2, x, oo, dir='-') = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(4 + 3/x**2, x, oo, dir='-') = 0 undefined where x = 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 4 incorrectly drops the 1/x**2 term from the numerator when applying the limit-law. The numerator should be Limit(4/x + 1/x**2, x, oo), not just Limit(4/x, x, oo).qwen3.6:27b-mlx: fail (error) — Step 4 incorrectly applies the quotient rule for limits by splitting the numerator into a single term `Limit(4/x, x, oo)` instead of the full numerator `Limit(4/x + 1/x**2, x, oo)`. This is a mathematical error in the application of the limit law.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 incorrectly applies the quotient rule for limits by splitting the numerator into a single term `Limit(4/x, x, oo)` instead of the full numerator `Limit(4/x + 1/x**2, x, oo)`. This is a mathematical error in the application of the limit law.gpt-oss:20b: fail (error) 2026-10-03 — Step 4 incorrectly drops the 1/x**2 term from the numerator when applying the limit-law. The numerator should be Limit(4/x + 1/x**2, x, oo), not just Limit(4/x, x, oo).qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 4 incorrectly applies the quotient rule for limits by splitting the numerator into two separate limit terms (Limit(4/x) / Limit(4 + 3/x**2)) instead of keeping the numerator as a single limit expression (Limit(4/x + 1/x**2)). This is a logical error in the application of limit laws.gpt-oss:20b: inconclusive 2026-09-30 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 4 incorrectly applies the limit-law to only the first term of the numerator, dropping the \(1/x^2\) term. The correct application would keep the enti
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.