Home›Calculus 1›Trigonometric limits›Problem 1.239

Limit of \( \displaystyle \frac{\tan{\left(3 x + 1 \right)}}{3 x + 1} \) as \( x \to - \frac{1}{3} \)

Problem 1.239 · medium

Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\tan{\left(3 x + 1 \right)}}{3 x + 1} \).

\[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\tan{\left(3 x + 1 \right)}}{3 x + 1}\right) \]
limit algebra simplifySet up the limit. Rewrite the denominator to isolate the term approaching zero. The expression is already in a form suitable for substitution.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\tan{\left(u \right)}}{u}\right) \]
substitutionLet u = 3*x + 1. As x approaches -1/3, u approaches 0.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \tan{\left(u \right)}}{\frac{d}{d u} u}\right) \]
lhopitalApply L'Hôpital's rule because the limit is of the form 0/0.✓ Proved
\[ = \lim_{u \to 0^+} \sec^{2}{\left(u \right)} \]
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = 1 \]
limitEvaluate the limit as u approaches 0.✓ 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 tan has poles at odd multiples of pi/2 undefined where 3*x + 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where 3*x + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where 3*x + 1 = 0 undefined where u = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where u = 0 undefined where Derivative(u, u) = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where Derivative(u, u) = 0 sec has poles at odd multiples of pi/2
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2
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: fail (error) — Step 3 reverts the expression from Step 2 back to the original form, undoing the algebraic manipulation in Step 2 without justification. This creates a logical discontinuity where the setup for the substitution in Step 4 is not derived from the previous line.

Senior review claude-sonnet-5-5, 2026-10-05: pass — Every line is a valid equality and the labels are legitimate. Steps 2 and 3 are a pointless round trip, which is only a stylistic blemish. The answer of 1 is correct.

qwen3.6:27b-mlx: dismiss — Step 3 is a true identity (3*(x+1/3) = 3x+1) that SymPy proved, and 'simplify' is a legitimate label for it; being redundant is not a defect under the contract. The Derivative(tan(u), u) notation in step 5 is correct for the substituted limit, and the 0/0 form is stated in that step's note.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 reverts the expression from Step 2 back to the original form, undoing the algebraic manipulation in Step 2 without justification. This creates a logical discontinuity where the setup for the substitution in Step 4 is not derived from the previous line.
gpt-oss:20b: pass 2026-10-05
claude-sonnet-5-5: pass 2026-10-05 — Every line is a valid equality and the labels are legitimate. Steps 2 and 3 are a pointless round trip, which is only a stylistic blemish. The answer of 1 is correct.
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 reverts the expression to its original form, undoing the algebraic manipulation in Step 2, which is logically redundant and confusing. Furthermore, Step 5 applies L'Hôpital's rule to the variable u, but the notation 'Derivative(tan(u), u)' implies differentiation with respect to u, which is correct for the substituted limit, yet the transition from Step 4 to 5 is abrupt without explicitly stating the 0/0 form check, though the main defect is the nonsensical Step 3 which adds no value and breaks the flow of simplification.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.