Limit of \( \displaystyle \frac{1 - \cos{\left(x \right)}}{x} \) as \( x \to 0 \)
Problem 1.33 · easy
Evaluate \( \displaystyle \lim_{x \to 0} \frac{1 - \cos{\left(x \right)}}{x} \).
- \[ \lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{x}\right) \]limitStarting with the original limit.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right)}{\frac{d}{d x} x}\right) \]lhopitalApply L'Hopital's rule since the limit is an indeterminate form 0/0.✓ Proved
- \[ = \lim_{x \to 0^+} \sin{\left(x \right)} \]simplifyCompute the derivatives of the numerator and denominator.✓ Proved
- \[ = 0 \]limitEvaluate the limit of sin(x) as x approaches 0.✓ Proved
Answer \( 0 \)
✓ 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 undefined where Derivative(x, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(x, x) = 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: passqwen3.6:27b-mlx: pass — The solution correctly applies L'Hopital's rule to the indeterminate form 0/0, computes the derivatives accurately, and evaluates the resulting limit. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies L'Hopital's rule to the indeterminate form 0/0, computes the derivatives accurately, and evaluates the resulting limit. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies L'Hopital's rule to the 0/0 indeterminate form and evaluates the resulting limit. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.