Home›Calculus 1›Trigonometric limits›Problem 1.33

Limit of \( \displaystyle \frac{1 - \cos{\left(x \right)}}{x} \) as \( x \to 0 \)

Problem 1.33 · easy Video

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 \( 0 \)

The graph of f(x) = (1 - cos(x))/x. Two points slide along the curve toward x = 0, one from each side; their heights close in on 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. 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 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies L'Hopital's rule to the 0/0 indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies L'Hopital's rule to the 0/0 indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-03
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 2026-09-29
qwen3.6:27b-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 2026-10-03 with SymPy 1.14.0.