Home›Calculus 1›Limits by factoring›Problem 1.11

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

Problem 1.11 · medium Video

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

\[ \lim_{x \to 1^+}\left(\frac{x^{3} - 1}{x^{2} - 1}\right) \]
limitStart with the original limit.✓ Proved
\[ = \lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(x^{2} + x + 1\right)}{x^{2} - 1}\right) \]
factorFactor the numerator using the difference of cubes formula.✓ Proved
\[ = \lim_{x \to 1^+}\left(\frac{x^{2} + x + 1}{x + 1}\right) \]
factor cancelFactor the denominator using the difference of squares formula. Cancel the common factor (x - 1).✓ Proved
\[ = \lim_{x \to 1^+}\left(x^{2} + x + 1\right) \left(\lim_{x \to 1^+}\left(x + 1\right)\right)^{-1} \]
limit-lawApply the quotient rule for limits.✓ Proved
\[ = \frac{3}{2} \]
limitEvaluate the limits of the numerator and denominator.✓ Proved

Answer \( \frac{3}{2} \)

The graph of f(x) = (x³ - 1)/(x² - 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 x**2 - 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 - 1 = 0 undefined where x + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where Limit(x + 1, x, 1, dir='+') = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Limit(x + 1, x, 1, dir='+') = 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 — The solution correctly applies factoring, cancellation, and limit laws in distinct steps. The labels are appropriate and the logic is sound.
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.