Limit of \( \displaystyle \frac{5 x - 7}{3 x + 4} \) as \( x \to \infty \)
Problem 1.242 · medium
- \[ \lim_{x \to \infty}\left(\frac{5 x - 7}{3 x + 4}\right) \]limitStart with the limit of the function as x approaches infinity.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{5 - \frac{7}{x}}{3 + \frac{4}{x}}\right) \]algebra simplifyDivide the numerator and denominator by x. Simplify the terms in the numerator and denominator.✓ Proved
- \[ = \lim_{x \to \infty}\left(5 - \frac{7}{x}\right) \left(\lim_{x \to \infty}\left(3 + \frac{4}{x}\right)\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = \lim_{x \to \infty} 5 \left(\lim_{x \to \infty} 3\right)^{-1} \]limitEvaluate the limits of the individual parts.✓ Proved
- \[ = \frac{5}{3} \]simplifyCalculate the final value.✓ Proved
✓ 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 3*x + 4 = 0 undefined where x = 0 undefined where 3 + 4/x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 3 + 4/x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 3 + 4/x = 0 undefined where Limit(3 + 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(3 + 4/x, x, oo, dir='-') = 0 undefined where Limit(3, x, oo, dir='-') = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(3, x, oo, dir='-') = 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
Senior review claude-sonnet-5-5, 2026-10-05: pass — The solution is correct and each step is a single named rule. Step 5 evaluates both sub-limits (7/x and 4/x go to 0), which is a legitimate use of the limit rule; step 6 then evaluates the constant limits and divides.
qwen3.6:27b-mlx: dismiss — Step 5 evaluates lim(5 - 7/x) as 5 and lim(3 + 4/x) as 3 in one application of the limit rule to each part (the contract treats applying one rule to several terms as a single change); 'limit' is a legitimate label, and SymPy verified the line.
Every verdict on record (5)
qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05claude-sonnet-5-5: pass 2026-10-05 — The solution is correct and each step is a single named rule. Step 5 evaluates both sub-limits (7/x and 4/x go to 0), which is a legitimate use of the limit rule; step 6 then evaluates the constant limits and divides.qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 5 claims to evaluate the limits of the individual parts, but it only removes the terms 7/x and 4/x (which go to 0) while leaving the limits of the constants 5 and 3 unevaluated. This violates the 'one change per step' rule by performing two distinct limit evaluations (or simplifications) simultaneously, and the label 'limit' is insufficiently specific for the partial evaluation performed.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.