Home›Calculus 1›Trigonometric limits›Problem 1.37

Limit of \( \displaystyle \frac{x}{\sin{\left(x \right)}} \) as \( x \to 0 \)

Problem 1.37 · easy Video

Evaluate \( \displaystyle \lim_{x \to 0} \frac{x}{\sin{\left(x \right)}} \).

\[ \lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) \]
limitIdentify the limit to be evaluated.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sin{\left(x \right)}}\right) \]
lhopitalThe limit is an indeterminate form 0/0, so apply L'Hôpital's rule.✓ Proved
\[ = \lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}} \]
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = 1 \]
limit simplifyEvaluate the limit by substituting x = 0. Simplify the final value.✓ Proved

Answer \( 1 \)

The graph of f(x) = x/sin(x). Two points slide along the curve toward x = 0, one from each side; their heights close in on 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. 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 sin(x) = 0 undefined where Derivative(sin(x), x) = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(sin(x), x) = 0 undefined where cos(x) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) = 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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29
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.