Home›Calculus 1›Limits at infinity›Problem 1.48

Limit of \( \displaystyle \frac{3 x^{2} + 1}{x^{2} - 4} \) as \( x \to \infty \)

Problem 1.48 · medium Video

Evaluate \( \displaystyle \lim_{x \to \infty} \frac{3 x^{2} + 1}{x^{2} - 4} \).

\[ \lim_{x \to \infty}\left(\frac{3 x^{2} + 1}{x^{2} - 4}\right) \]
limitStart with the original limit expression.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{3 + \frac{1}{x^{2}}}{1 - \frac{4}{x^{2}}}\right) \]
algebra simplifyDivide both numerator and denominator by the highest power of x, which is x**2. Simplify the fractions within the expression.✓ Proved
\[ = \lim_{x \to \infty}\left(3 + \frac{1}{x^{2}}\right) \left(\lim_{x \to \infty}\left(1 - \frac{4}{x^{2}}\right)\right)^{-1} \]
limit-lawApply the quotient rule for limits.✓ Proved
\[ = 3 \]
limit simplifyEvaluate the limits of the numerator and denominator separately as x approaches infinity. Final result.✓ Proved

Answer \( 3 \)

The graph of f(x) = (3x² + 1)/(x² - 4). A point runs right along the curve; its height settles toward 3.

✓ 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 x2 - 4 = 0 undefined where 1 - 4/x2 = 0 undefined where x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - 4/x**2 = 0 undefined where x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - 4/x2 = 0 undefined where x = 0 undefined where Limit(1 - 4/x2, x, oo, dir='-') = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where Limit(1 - 4/x**2, x, oo, dir='-') = 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 5 applies the limit to both the numerator and denominator simultaneously, violating the 'one change per step' rule. It should first evaluate the numerator limit, then the denominator limit in separate steps.

Senior review claude-sonnet-5-5, 2026-10-03: pass — The solution is correct: divide by x**2, simplify, apply the quotient limit law, evaluate the limits (3 and 1), and get 3. Rule labels are all legitimate and every line was proved by SymPy.

qwen3.6:27b-mlx: dismiss — Evaluating the numerator and denominator limits in one step is one application of the 'limit' rule to two terms, which the contract counts as a single change; the line is mathematically correct, with SymPy confirming equality.

Every verdict on record (7)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 applies the limit to both the numerator and denominator simultaneously, violating the 'one change per step' rule. It should first evaluate the numerator limit, then the denominator limit in separate steps.
gpt-oss:20b: pass 2026-10-03
claude-sonnet-5-5: pass 2026-10-03 — The solution is correct: divide by x**2, simplify, apply the quotient limit law, evaluate the limits (3 and 1), and get 3. Rule labels are all legitimate and every line was proved by SymPy.
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 claims to evaluate the limits of the numerator and denominator separately but only provides the final constants (3 and 1) without showing the intermediate limit evaluations (e.g., Limit(3 + 1/x**2, x, oo) -> 3). This violates the 'one thing per step' rule by combining the evaluation of two separate limits and the application of limit laws into a single jump.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 applies the limit to both the numerator and denominator simultaneously, violating the one-change-per-step rule. It should be split into two steps, one for the numerator and one for the denominator.
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.