Home›Calculus 1›Limits by factoring›Problem 1.703

Limit of $ \displaystyle \frac{\left(2 x - 1\right)^{3} - 1}{\left(2 x - 1\right)^{2} - 1} $ as $ x \to 1 $

Problem 1.703 · medium Video

Evaluate $ \displaystyle \lim_{x \to 1} \frac{\left(2 x - 1\right)^{3} - 1}{\left(2 x - 1\right)^{2} - 1} $.

\[ \lim_{x \to 1^+}\left(\frac{\left(2 x - 1\right)^{3} - 1}{\left(2 x - 1\right)^{2} - 1}\right) \]
limitStart with the limit of the original function.✓ Proved
\[ = \lim_{x \to 1^+}\left(\frac{\left(2 x - 2\right) \left(2 x + \left(2 x - 1\right)^{2}\right)}{\left(2 x - 1\right)^{2} - 1}\right) \]
factorFactor the numerator using the difference of cubes formula $a^3 - b^3 = (a-b)(a^2+ab+b^2)$.✓ Proved
\[ = \lim_{x \to 1^+}\left(\frac{2 x + \left(2 x - 1\right)^{2}}{2 x}\right) \]
factor cancel algebraFactor the denominator using the difference of squares formula $a^2 - b^2 = (a-b)(a+b)$. Cancel the common factor $(2x - 2)$ from the numerator and denominator. Simplify the denominator.✓ Proved
\[ = \lim_{x \to 1^+} \frac{3}{2} \]
substitutionSubstitute $x = 1$ into the expression.✓ Proved
\[ = \frac{3}{2} \]
limitEvaluate the final numerical value.✓ Proved

Answer $ \frac{3}{2} $

The graph of f(x) = ((2x - 1)³ - 1)/((2x - 1)² - 1). Two points slide along the curve toward x = 1, one from each side; their heights close in on 3/2.

✓ 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 (2x - 1)*2 - 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (2x - 1)*2 - 1 = 0 undefined where x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
7	✓ 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 factoring, cancellation, and substitution rules. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies factoring, cancellation, and substitution rules. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly factors the numerator and denominator, cancels the common term, simplifies, and substitutes to find the limit. Each step applies a single rule and is correctly labeled.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies factoring, cancellation, and substitution rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.

Limit of \( \displaystyle \frac{\left(2 x - 1\right)^{3} - 1}{\left(2 x - 1\right)^{2} - 1} \) as \( x \to 1 \)