Limit of \( \displaystyle \frac{\cos{\left(x - 1 \right)}}{x} \) as \( x \to 1 \)
Problem 1.118 · easy
Evaluate \( \displaystyle \lim_{x \to 1} \frac{\cos{\left(x - 1 \right)}}{x} \).
- \[ \lim_{x \to 1^+}\left(\frac{\cos{\left(x - 1 \right)}}{x}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = 1 \]limit simplify simplify simplifySince the denominator is non-zero, we can evaluate the limit by direct substitution. Simplify the argument of the cosine function. Evaluate the cosine of zero. Final simplification.✓ Proved
Answer \( 1 \)
✓ 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 x = 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 |
| 5 | ✓ 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 direct substitution for a continuous function at a point where the denominator is non-zero. Each step changes only one thing and uses valid labels.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies direct substitution for a continuous function at a point where the denominator is non-zero. Each step changes only one thing and uses valid labels.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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.