Limit of \( \displaystyle \frac{15 x - 2}{9 x + 7} \) as \( x \to \infty \)
Problem 1.95 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{15 x - 2}{9 x + 7} \).
- \[ \lim_{x \to \infty}\left(\frac{15 x - 2}{9 x + 7}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{15 - \frac{2}{x}}{9 + \frac{7}{x}}\right) \]algebra simplifyDivide both the numerator and the denominator by x. Simplify the terms inside the limit.✓ Proved
- \[ = \lim_{x \to \infty}\left(15 - \frac{2}{x}\right) \left(\lim_{x \to \infty}\left(9 + \frac{7}{x}\right)\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = \frac{5}{3} \]limit simplifyEvaluate the limits of the numerator and denominator separately as x approaches infinity. Simplify the resulting fraction.✓ Proved
Answer \( \frac{5}{3} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 + 7 = 0 undefined where 9 + 7/x = 0 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9 + 7/x = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9 + 7/x = 0 undefined where x = 0 undefined where Limit(9 + 7/x, x, oo, dir='-') = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(9 + 7/x, 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: passqwen3.6:27b-mlx: pass — The solution correctly applies algebraic manipulation, limit laws, and simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Senior review claude-sonnet-5-5, 2026-10-03: pass — Every line is correct and SymPy proved each one. The rule names fit, and evaluating both limits in step 5 is one application of the limit rule. The only quibble is that step 5 does a little extra, which is not a defect.
qwen3.6:27b-mlx: dismiss — Line 5 applies the single rule 'limit' to both factors of the quotient, which counts as one change under the contract. Writing 15/9 is just the quotient of the two evaluated limits, and 15/9 is correct (SymPy checked).
Every verdict on record (5)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies algebraic manipulation, limit laws, and simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03claude-sonnet-5-5: pass 2026-10-03 — Every line is correct and SymPy proved each one. The rule names fit, and evaluating both limits in step 5 is one application of the limit rule. The only quibble is that step 5 does a little extra, which is not a defect.qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 5 is mathematically incorrect because it evaluates the limit of the numerator (15 - 2/x) as 15 and the denominator (9 + 7/x) as 9, but writes the result as 15/9 without showing the intermediate step of evaluating the limits of the individual terms (15 and 9). More critically, the transition from Step 4 to Step 5 skips the actual evaluation of the limits `Limit(15 - 2/x, x, oo)` -> `15` and `Limit(9 + 7/x, x, oo)` -> `9`. The label 'limit' is applied to a step that performs arithmetic division on the results of limits, but the step itself jumps from the limit expressions to the final fraction 15/9, effectively combining the evaluation of the limits and the division into one step, or implying that `Limit(15 - 2/x, x, oo)` is directly `15` and `Limit(9 + 7/x, x, oo)` is directly `9` in a single jump. While often accepted in shorthand, strictly speaking, Step 5 should be split or labeled more carefully if it's doing two things. However, the bigger issue is that Step 5 claims the result is `15/9` directly from the limit expressions. It misses the intermediate step of evaluating the limits to 15 and 9 respectively. Actually, looking closer, Step 5 says `15 / 9`. This is the result of evaluating the limits. The rule `limit` is used. Is it one rule? It evaluates two limits and divides. That's multiple operations. But the contract says 'Each step changes ONE thing'. Step 5 changes `Limit(15 - 2/x, x, oo)` to `15` AND `Limit(9 + 7/x, x, oo)` to `9` AND performs the division. That is three things. It should be split.gpt-oss:20b: pass 2026-09-30
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.