Home›Calculus 1›Limits by direct substitution›Problem 1.17

Limit of \( \displaystyle 4 x^{2} + 6 x - 1 \) as \( x \to 1 \)

Problem 1.17 · medium Video

Evaluate \( \displaystyle \lim_{x \to 1} 4 x^{2} + 6 x - 1 \).

\[ \lim_{x \to 1^+}\left(4 x^{2} + 6 x - 1\right) \]
limitIdentify the limit to be evaluated.✓ Proved
\[ = 9 \]
limitSince the function is a polynomial, it is continuous, so we can evaluate by direct substitution.✓ Proved

Answer \( 9 \)

The graph of f(x) = 4x² + 6x - 1. Two points slide along the curve toward x = 1, one from each side; their heights close in on 9.

✓ 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
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: fail (style) — The second step applies a direct substitution into the polynomial, which should be labeled with the rule "substitution" (or "simplify"), not "limit". The label does not match the rule actually used.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03
gpt-oss:20b: fail (style) 2026-10-03 — The second step applies a direct substitution into the polynomial, which should be labeled with the rule "substitution" (or "simplify"), not "limit". The label does not match the rule actually used.
qwen3.6:27b-mlx: pass 2026-09-29
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.