Limit of \( \displaystyle \frac{10 x - 7}{6 x + 4} \) as \( x \to \infty \)
Problem 1.255 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{10 x - 7}{6 x + 4} \).
- \[ \lim_{x \to \infty}\left(\frac{10 x - 7}{6 x + 4}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{10 - \frac{7}{x}}{6 + \frac{4}{x}}\right) \]algebra simplifyDivide the numerator and denominator by x. Simplify the terms inside the limit.✓ Proved
- \[ = \lim_{x \to \infty} 10 \left(\lim_{x \to \infty}\left(6 + \frac{4}{x}\right)\right)^{-1} \]limit-lawApply the limit law for the quotient of the numerator and denominator.✓ Proved
- \[ = \lim_{x \to \infty} 10 \left(\lim_{x \to \infty} 6\right)^{-1} \]limit-lawEvaluate the limit of the denominator.✓ Proved
- \[ = \frac{5}{3} \]limit simplifyEvaluate the remaining limits. Simplify the final fraction.✓ Proved
Answer \( \frac{5}{3} \)
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. The reviewers disagree about how one step is explained; every verdict is in 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 6*x + 4 = 0 undefined where 6 + 4/x = 0 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 6 + 4/x = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 6 + 4/x = 0 undefined where x = 0 undefined where Limit(6 + 4/x, x, oo, dir='-') = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where Limit(6 + 4/x, x, oo, dir='-') = 0 undefined where Limit(6, x, oo, dir='-') = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(6, x, oo, dir='-') = 0 |
| 7 | ✓ 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 applies the limit law: it replaces the limit of the quotient \(\lim\frac{10-7/x}{6+4/x}\) with \(\frac{\lim 10}{\lim(6+4/x)}\). The numerator limit \(\lim(10-7/x)\) is missing, so the step applies more than one rule at once and is mathematically incorrect.qwen3.6:27b-mlx: fail (error) — Step 5 is mathematically incorrect; it replaces Limit(6 + 4/x, x, oo) with Limit(6, x, oo) without showing the evaluation of the 4/x term or applying limit laws to split the sum. The transition skips the necessary step of evaluating Limit(4/x, x, oo) = 0.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 5 is mathematically incorrect; it replaces Limit(6 + 4/x, x, oo) with Limit(6, x, oo) without showing the evaluation of the 4/x term or applying limit laws to split the sum. The transition skips the necessary step of evaluating Limit(4/x, x, oo) = 0.gpt-oss:20b: fail (error) 2026-10-05 — Step 4 incorrectly applies the limit law: it replaces the limit of the quotient \(\lim\frac{10-7/x}{6+4/x}\) with \(\frac{\lim 10}{\lim(6+4/x)}\). The numerator limit \(\lim(10-7/x)\) is missing, so the step applies more than one rule at once and is mathematically incorrect.qwen3.6:27b-mlx: fail (style) 2026-10-05 — Step 5 applies two rules at once: it uses the limit law for sums to split the denominator and then evaluates the limit of the constant term 6, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.