Limit of \( \displaystyle \frac{\ln{\left(3 x \right)}}{3 x} \) as \( x \to \infty \)
Problem 1.109 · easy
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{\ln{\left(3 x \right)}}{3 x} \).
- \[ \lim_{x \to \infty}\left(\frac{\ln{\left(3 x \right)}}{3 x}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{\ln{\left(3 \right)}}{3 x^{2}}\right) \]simplify algebra algebraRewrite the expression using power notation. Combine the terms in the numerator and denominator. Separate the constant coefficient.✓ Proved
- \[ = 0 \]limitAs x approaches infinity, 1/x^2 approaches 0.✓ Proved
Answer \( 0 \)
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. A reviewer's objection to the explanation stands; see 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 log is undefined for non-positive arguments undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 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 2 incorrectly rewrites log(3*x) as log(3)*x**(-1); the correct expansion is log(3)+log(x). Subsequent algebraic manipulations are based on this incorrect form, leading to an invalid conclusion.qwen3.6:27b-mlx: fail (error) — Step 2 incorrectly transforms log(3*x) into log(3)*x, which is mathematically false. The original expression is log(3x)/(3x), but the solver treats it as if the numerator were log(3)*x.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 incorrectly transforms log(3*x) into log(3)*x, which is mathematically false. The original expression is log(3x)/(3x), but the solver treats it as if the numerator were log(3)*x.gpt-oss:20b: fail (error) 2026-10-03 — Step 2 incorrectly rewrites log(3*x) as log(3)*x**(-1); the correct expansion is log(3)+log(x). Subsequent algebraic manipulations are based on this incorrect form, leading to an invalid conclusion.qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 incorrectly transforms log(3*x) into log(3)*x, which is mathematically false. The expression log(3*x) cannot be simplified to log(3)*x; this is a fundamental algebraic error that changes the function being evaluated.gpt-oss:20b: fail (error) 2026-10-03 — Step 2 incorrectly rewrites log(3*x) as log(3)*x**(-1); the correct expansion is log(3)+log(x). Subsequent steps depend on this incorrect form, so the solution is mathematically invalid.
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.