Limit of \( \displaystyle \frac{10 x + 3}{6 x + 10} \) as \( x \to \infty \)
Problem 1.370 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{10 x + 3}{6 x + 10} \).
- \[ \lim_{x \to \infty}\left(\frac{10 x + 3}{6 x + 10}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{10 + \frac{3}{x}}{6 + \frac{10}{x}}\right) \]algebra simplifyDivide the numerator and denominator by x. Simplify the terms inside the limit.✓ Proved
- \[ = \lim_{x \to \infty} 10 \left(\lim_{x \to \infty} 6\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = \frac{5}{3} \]limit simplifyEvaluate the limits of the constant terms. Reduce the fraction to its simplest form.✓ 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 + 10 = 0 undefined where x = 0 undefined where 6 + 10/x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 6 + 10/x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 6 + 10/x = 0 undefined where Limit(6, x, oo, dir='-') = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(6, x, oo, dir='-') = 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 applies the limit law by discarding the 3/x and 10/x terms; the limit of (10+3/x)/(6+10/x) should be taken as the quotient of the limits of the entire numerator and denominator, not just the constant parts.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 4 incorrectly applies the limit law by discarding the 3/x and 10/x terms; the limit of (10+3/x)/(6+10/x) should be taken as the quotient of the limits of the entire numerator and denominator, not just the constant parts.qwen3.6:27b-mlx: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.