Home›Calculus 1›Limits at infinity›Problem 1.19

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

Problem 1.19 · hard Video

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

\[ \lim_{x \to \infty}\left(\frac{4 \left(x + 1\right)^{2} - 1}{x + 2 \left(x + 1\right)^{2} + 1}\right) \]
limitEvaluate the limit as x approaches infinity.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{4 x^{2} + 8 x + 3}{x + 2 \left(x + 1\right)^{2} + 1}\right) \]
algebraExpand the squared term in the numerator.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{4 x^{2} + 8 x + 3}{2 x^{2} + 5 x + 3}\right) \]
algebra algebra simplifyDistribute the 4 in the numerator. Expand the term in the denominator. Combine like terms in the denominator.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{4 + \frac{8}{x} + \frac{3}{x^{2}}}{2 + \frac{5}{x} + \frac{3}{x^{2}}}\right) \]
algebra simplifyDivide both numerator and denominator by x**2. Simplify the fractions.✓ Proved
\[ = \lim_{x \to \infty} 2 \]
limit-lawApply the limit laws for the terms approaching zero.✓ Proved
\[ = 2 \]
limitEvaluate the final value.✓ Proved

Answer \( 2 \)

The graph of f(x) = (4(x + 1)² - 1)/(x + 2(x + 1)² + 1). A point runs right along the curve; its height settles toward 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(x + 1)*2 + 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 2(x + 1)2 + 1 = 0 undefined where 2x*2 + 5x + 3 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2x2 + 5x + 3 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2x2 + 5x + 3 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2x2 + 5x + 3 = 0 undefined where 2 + 5/x + 3/x**2 = 0 undefined where x = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2 + 5/x + 3/x**2 = 0 undefined where x = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2 + 5/x + 3/x**2 = 0 undefined where x = 0
9	✓ 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 (error) — Step 8 applies the limit to multiple terms simultaneously (8/x, 3/x^2, 5/x, 3/x^2) and simplifies the constants in one step, violating the 'one change per step' rule. It should have been broken down into separate steps for each term or applied via limit laws step-by-step.

Senior review claude-sonnet-5-5, 2026-10-04: pass — The solution is correct: the limit of (4x^2+8x+3)/(2x^2+5x+3) is 4/2 = 2. Each step is labeled with a rule from the allowed vocabulary and every line was proved by SymPy.

qwen3.6:27b-mlx: dismiss — Applying the limit laws to the terms that vanish as x->oo is one limit-law step (one rule applied across the terms), and the contract allows this; line 8 is correct and SymPy proved it.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-04 — The solution is correct: the limit of (4x^2+8x+3)/(2x^2+5x+3) is 4/2 = 2. Each step is labeled with a rule from the allowed vocabulary and every line was proved by SymPy.
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 8 applies the limit to multiple terms simultaneously (8/x, 3/x^2, 5/x, 3/x^2) and simplifies the constants in one step, violating the 'one change per step' rule. It should have been broken down into separate steps for each term or applied via limit laws step-by-step.
gpt-oss:20b: pass 2026-10-03
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.