Home›Calculus 1›Limits at infinity›Problem 1.79

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

Problem 1.79 · medium

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

\[ \lim_{x \to \infty}\left(\frac{\sqrt{\left(3 x + 1\right)^{2} + 1}}{3 x + 1}\right) \]
limitStart with the original limit expression.✓ Proved
\[ = \lim_{x \to \infty} \sqrt{1 + \frac{1}{\left(3 x + 1\right)^{2}}} \]
algebra simplifyDivide the numerator and denominator by (3*x + 1). Simplify the expression inside the square root.✓ Proved
\[ = \lim_{x \to \infty}\left(1 + \frac{1}{\left(3 x + 1\right)^{2}}\right) \]
limit-lawApply the limit law for the square root function.✓ Proved
\[ = \lim_{x \to \infty} 1 \]
infinityThe term 1/(3*x + 1)**2 approaches 0 as x approaches infinity.✓ Proved
\[ = 1 \]
limitEvaluate the final limit.✓ Proved

Answer \( 1 \)

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 3*x + 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x + 1 = 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: fail (error) — Step 4 removes the square‑root incorrectly; the limit of sqrt(1+… ) is sqrt(limit(1+…)), not 1+…
qwen3.6:27b-mlx: pass — The solution correctly applies algebraic manipulation to simplify the expression inside the limit, followed by standard limit laws and evaluation at infinity. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.

Senior review claude-sonnet-5-5, 2026-10-03: fail — Lines 2 and 3 are sound. Line 4 drops the sqrt while citing the square-root limit law, which would give sqrt(Limit(1+1/(3x+1)^2)). A student could come away thinking lim sqrt(f) equals lim f. The final answer of 1 is correct.

gpt-oss:20b: uphold — The step 2 claim is wrong, since moving (3x+1) under the root as (3x+1)^2 is valid for x > -1/3 and SymPy proved it. The step 4 claim is right: the limit-law step drops the square root, so line 4 is Limit(inner) rather than sqrt(Limit(inner)). That is not what the named law gives, and it hides the sqrt(1)=1 evaluation.

Every verdict on record (5)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies algebraic manipulation to simplify the expression inside the limit, followed by standard limit laws and evaluation at infinity. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
gpt-oss:20b: fail (error) 2026-10-03 — Step 4 removes the square‑root incorrectly; the limit of sqrt(1+… ) is sqrt(limit(1+…)), not 1+…
claude-sonnet-5-5: fail (misleading) 2026-10-03 — Lines 2 and 3 are sound. Line 4 drops the sqrt while citing the square-root limit law, which would give sqrt(Limit(1+1/(3x+1)^2)). A student could come away thinking lim sqrt(f) equals lim f. The final answer of 1 is correct.
qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies algebraic manipulation to simplify the expression inside the limit, followed by standard limit laws and evaluation. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.
gpt-oss:20b: fail (error) 2026-09-30 — Step 2 incorrectly rewrites the expression by dividing terms inside the square root; this applies two transformations at once and is not algebraically valid. Step 4 then misapplies a limit law for the square root. The solution contains a fundamental algebraic error.

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.