Home›Calculus 1›Limits by factoring›Problem 1.25

Limit of \( \displaystyle \frac{- x + \left(x + 1\right)^{2} - 1}{\left(x + 1\right)^{2} - 1} \) as \( x \to 0 \)

Problem 1.25 · medium Video

Evaluate \( \displaystyle \lim_{x \to 0} \frac{- x + \left(x + 1\right)^{2} - 1}{\left(x + 1\right)^{2} - 1} \).

\[ \lim_{x \to 0^+}\left(\frac{- x + \left(x + 1\right)^{2} - 1}{\left(x + 1\right)^{2} - 1}\right) \]
limitStart with the limit of the given function.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{x^{2} + x}{x^{2} + 2 x}\right) \]
algebra simplify simplifyExpand the squared terms. Simplify the numerator and denominator. Combine like terms in the numerator.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{x + 1}{x + 2}\right) \]
factor cancelFactor out x from both numerator and denominator. Cancel the common factor x.✓ Proved
\[ = 0.5 \]
limitEvaluate the limit by substituting x = 0.✓ Proved

Answer \( \frac{1}{2} \)

The graph of f(x) = (-x + (x + 1)² - 1)/((x + 1)² - 1). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/2.

✓ 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 + 1)2 - 1 = 0 undefined where x2 + 2*x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 2x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 2x = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 2x = 0 undefined where x + 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 2 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 2 = 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 algebraic simplification, factoring, and cancellation to resolve the indeterminate form, followed by direct substitution. Each step changes only one aspect of the expression and uses valid labels.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form, followed by direct substitution. Each step changes only one aspect of the expression and uses valid labels.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form, followed by direct substitution. Each step changes only one aspect of the expression and uses valid labels.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form, followed by direct substitution. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
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.