Home›Calculus 1›Limits at infinity›Problem 1.60

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

Problem 1.60 · medium

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

\[ \lim_{x \to \infty}\left(\frac{2 x + 1}{x^{2} + 3}\right) \]
limitStart with the original limit expression.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{\frac{2}{x} + \frac{1}{x^{2}}}{1 + \frac{3}{x^{2}}}\right) \]
algebra simplifyDivide the numerator and denominator by the highest power of x in the denominator, which is x**2. Simplify the terms inside the limit.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{2}{x}\right) \left(\lim_{x \to \infty}\left(1 + \frac{3}{x^{2}}\right)\right)^{-1} \]
limit-lawApply the quotient rule for limits.✓ Proved
\[ = 0 \]
limit simplifyEvaluate the limits of the individual components as x approaches infinity. Final simplification.✓ Proved

Answer \( 0 \)

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. A reviewer's objection to the explanation stands; see the receipt.

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 x2 + 3 = 0 undefined where 1 + 3/x2 = 0 undefined where x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 + 3/x**2 = 0 undefined where x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 + 3/x2 = 0 undefined where x = 0 undefined where Limit(1 + 3/x2, x, oo, dir='-') = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Limit(1 + 3/x**2, x, oo, dir='-') = 0 undefined where x = 0
6	✓ 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 4 applies the quotient rule for limits, but the numerator is not simplified to a single limit term (Limit(2/x + 1/x**2)) before the split. The step implicitly applies the sum rule for limits to the numerator and the quotient rule simultaneously, violating the one-rule-per-step constraint.

Senior review claude-sonnet-5-5, 2026-10-03: fail — Step 4 is labeled limit-law (quotient rule), but it discards the 1/x**2 term from the numerator. A correct step would write Limit(2/x + 1/x**2, x, oo) / Limit(1 + 3/x**2, x, oo) and then use the sum rule. The final answer 0 is correct, but a student would learn that terms can be dropped inside a limit.

gpt-oss:20b: uphold — Line 3's numerator is 2/x + 1/x**2, but line 4 keeps only Limit(2/x), so the quotient rule was not applied faithfully. The 1/x**2 term was silently dropped, which is a second change in the same step and teaches a false limit-law move, even though both versions evaluate to 0.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 applies the quotient rule for limits, but the numerator is not simplified to a single limit term (Limit(2/x + 1/x**2)) before the split. The step implicitly applies the sum rule for limits to the numerator and the quotient rule simultaneously, violating the one-rule-per-step constraint.
gpt-oss:20b: pass 2026-10-03
claude-sonnet-5-5: fail (misleading) 2026-10-03 — Step 4 is labeled limit-law (quotient rule), but it discards the 1/x**2 term from the numerator. A correct step would write Limit(2/x + 1/x**2, x, oo) / Limit(1 + 3/x**2, x, oo) and then use the sum rule. The final answer 0 is correct, but a student would learn that terms can be dropped inside a limit.
qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies algebraic manipulation, limit laws, and evaluation. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: fail (error) 2026-09-30 — Step 4 incorrectly applies the limit-law: it drops the “1/x**2” term from the numerator, treating the limit of the whole numerator as just the limit of 2/x. This violates the rule that each step must change only one thing and correctly apply the limit properties.

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.