Home›Calculus 1›Limits by factoring›Problem 1.558

Limit of \( \displaystyle \frac{- 10 x + \left(2 x - 1\right)^{2} + 11}{2 x - 4} \) as \( x \to 2 \)

Problem 1.558 · medium

Evaluate \( \displaystyle \lim_{x \to 2} \frac{- 10 x + \left(2 x - 1\right)^{2} + 11}{2 x - 4} \).

\[ \lim_{x \to 2^+}\left(\frac{- 10 x + \left(2 x - 1\right)^{2} + 11}{2 x - 4}\right) \]
limitStart with the limit of the given function.✓ Proved
\[ = \lim_{x \to 2^+}\left(\frac{4 x^{2} - 14 x + 12}{2 x - 4}\right) \]
simplify factorExpand and combine like terms. Factor out the common factor 2.✓ Proved
\[ = \lim_{x \to 2^+}\left(\frac{\left(x - 2\right) \left(4 x - 6\right)}{2 x - 4}\right) \]
factorFactor the quadratic expression.✓ Proved
\[ = \lim_{x \to 2^+}\left(2 x - 3\right) \]
cancelCancel the common factor (x - 2).✓ Proved
\[ = 1 \]
limitEvaluate the limit.✓ 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 2*x - 4 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x - 4 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x - 4 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x - 4 = 0
6	✓ 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: fail (style) — Step 5 applies two rules at once: it cancels the factor (x-2) AND cancels the constant factor 2. The step should be split into a cancellation of (x-2) followed by a simplification/cancellation of the constants, or the label should reflect that multiple operations occurred, but the contract requires one rule per step.

Senior review claude-sonnet-5-5, 2026-10-12: pass — All lines are algebraically correct and the rule labels are legitimate; the limit equals 1.

qwen3.6:27b-mlx: dismiss — Cancelling the common factor 2(x-2) is a single cancel step; the constant 2 is part of the same common factor, so only one change is made.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (style) 2026-10-12 — Step 5 applies two rules at once: it cancels the factor (x-2) AND cancels the constant factor 2. The step should be split into a cancellation of (x-2) followed by a simplification/cancellation of the constants, or the label should reflect that multiple operations occurred, but the contract requires one rule per step.
claude-sonnet-5-5: pass 2026-10-12 — All lines are algebraically correct and the rule labels are legitimate; the limit equals 1.
qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 5 applies two rules at once: it cancels the factor (x-2) and also cancels the constant factor 2 in the numerator and denominator. The 'cancel' rule should only remove the (x-2) term, leaving the constant 2, which would then require a subsequent simplification step to reach 2*x - 3.
gpt-oss:20b: pass 2026-10-12

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 2026-10-12 with SymPy 1.14.0.