Limit of \( \displaystyle \frac{\cos{\left(3 x + 1 \right)} - 1}{e^{3 x + 1} - 1} \) as \( x \to - \frac{1}{3} \)
Problem 1.750 · medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\cos{\left(3 x + 1 \right)} - 1}{e^{3 x + 1} - 1} \).
- \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\cos{\left(3 x + 1 \right)} - 1}{e^{3 x + 1} - 1}\right) \]limitIdentify the limit and the indeterminate form.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(\cos{\left(3 x + 1 \right)} - 1\right)}{\frac{d}{d x} \left(e^{3 x + 1} - 1\right)}\right) \]lhopitalApply L'Hopital's rule because the limit is in the form 0/0.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(- e^{- 3 x - 1} \sin{\left(3 x + 1 \right)}\right) \]simplify simplifyCalculate the derivatives and simplify the expression. Cancel the common factor of 3.✓ Proved
- \[ = \lim_{u \to 0^+}\left(- e^{- u} \sin{\left(u \right)}\right) \]substitutionUse substitution u = 3*x + 1, so as x approaches -1/3, u approaches 0.✓ Proved
- \[ = 0 \]limit simplify simplifyEvaluate the limit by plugging in the value of u. Simplify the resulting expression. Final result.✓ 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. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
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 exp(3*x + 1) - 1 = 0 undefined where Derivative(exp(3*x + 1) - 1, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(exp(3*x + 1) - 1, x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ 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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-25gpt-oss:20b: pass 2026-10-25qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 3 violates the 'one rule per step' constraint by applying differentiation (derivative) and algebraic simplification (canceling the factor of 3) simultaneously. The step should be split into a derivative step and a separate simplification step.gpt-oss:20b: pass 2026-10-24qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 3 applies two rules at once: it computes the derivatives (derivative rule) and simplifies the resulting expression (simplify rule) in a single step. The contract requires each step to change only one thing.gpt-oss:20b: pass 2026-10-24
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 2026-10-25 with SymPy 1.14.0.