Limit of \( \displaystyle \frac{15 x + 3}{9 x + 10} \) as \( x \to \infty \)
Problem 1.168 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{15 x + 3}{9 x + 10} \).
- \[ \lim_{x \to \infty}\left(\frac{15 x + 3}{9 x + 10}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{15 + \frac{3}{x}}{9 + \frac{10}{x}}\right) \]algebra cancelFactor out x from the numerator and denominator. Cancel the common factor of x.✓ Proved
- \[ = \lim_{x \to \infty}\left(15 + \frac{3}{x}\right) \left(\lim_{x \to \infty}\left(9 + \frac{10}{x}\right)\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = \lim_{x \to \infty} 15 \left(\lim_{x \to \infty} 9\right)^{-1} \]limit-lawEvaluate the limits of the individual terms.✓ Proved
- \[ = \frac{5}{3} \]limit simplifyEvaluate the resulting fraction. Simplify the fraction to its lowest terms.✓ 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 9*x + 10 = 0 undefined where 9 + 10/x = 0 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9 + 10/x = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9 + 10/x = 0 undefined where x = 0 undefined where Limit(9 + 10/x, x, oo, dir='-') = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(9 + 10/x, x, oo, dir='-') = 0 undefined where x = 0 undefined where Limit(9, x, oo, dir='-') = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(9, 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: passqwen3.6:27b-mlx: fail (error) — Step 5 is mathematically incorrect. The step claims that Limit(15 + 3/x, x, oo) equals Limit(15, x, oo), which implies that the limit of 3/x is zero, but it does not explicitly evaluate or remove the 3/x term. A correct step would either evaluate the limit of the constant term and the vanishing term separately (e.g., 15 + 0) or use a limit law to split the sum before evaluating. As written, it asserts an equality between a limit containing a variable term and a limit of a constant without showing the variable term vanishes, which is a logical gap/error in the derivation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 5 is mathematically incorrect. The step claims that Limit(15 + 3/x, x, oo) equals Limit(15, x, oo), which implies that the limit of 3/x is zero, but it does not explicitly evaluate or remove the 3/x term. A correct step would either evaluate the limit of the constant term and the vanishing term separately (e.g., 15 + 0) or use a limit law to split the sum before evaluating. As written, it asserts an equality between a limit containing a variable term and a limit of a constant without showing the variable term vanishes, which is a logical gap/error in the derivation.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies limit laws and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.