Limit of \( \displaystyle \frac{\cos{\left(3 x \right)}}{3 x + 1} \) as \( x \to 0 \)
Problem 1.86 · medium
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\cos{\left(3 x \right)}}{3 x + 1} \).
- \[ \lim_{x \to 0^+}\left(\frac{\cos{\left(3 x \right)}}{3 x + 1}\right) \]limitStart with the given limit.✓ Proved
- \[ = \lim_{x \to 0^+} 1 \]substitution simplifySubstitute x = 0 into the expression. Evaluate the constants in the expression.✓ Proved
- \[ = 1 \]limitThe limit evaluates to 1.✓ Proved
Answer \( 1 \)
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 3*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ 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 2 incorrectly substitutes x=0 inside the limit expression. The limit of a function at a point cannot be evaluated by simply plugging in the point unless the function is known to be continuous there. This step applies an invalid rule and therefore the solution is mathematically incorrect.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — Step 2 incorrectly substitutes x=0 inside the limit expression. The limit of a function at a point cannot be evaluated by simply plugging in the point unless the function is known to be continuous there. This step applies an invalid rule and therefore the solution is mathematically incorrect.qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies direct substitution for a continuous function at x=0. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.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.