Home›Calculus 1›Limits at infinity›Problem 1.459

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

Problem 1.459 · medium

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

\[ \lim_{x \to \infty}\left(\frac{x + 2}{\sqrt{\left(x + 1\right)^{2} + 4}}\right) \]
limitStart with the original limit expression.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{x + 2}{\sqrt{x^{2} + 2 x + 5}}\right) \]
algebra simplifyExpand the expression inside the square root. Combine like terms inside the square root.✓ Proved
\[ = \lim_{x \to \infty} \frac{1}{\sqrt{\frac{x^{2} + 2 x + 5}{x^{2}}}} \]
rewrite simplifyDivide the numerator and denominator by x (noting x > 0 as x approaches infinity). Simplify the numerator.✓ Proved
\[ = \lim_{x \to \infty} \frac{1}{\sqrt{1 + \frac{2}{x} + \frac{5}{x^{2}}}} \]
simplifyDistribute the division by x^2 inside the square root.✓ Proved
\[ = \lim_{x \to \infty} 1 \]
limitEvaluate the limit of the terms involving 1/x as x approaches infinity.✓ Proved
\[ = 1 \]
limitThe limit evaluates to 1.✓ 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 (x + 1)2 + 4 = 0 undefined where x2 + 2*x + 5 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 2x + 5 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 2x + 5 = 0 undefined where x = 0 undefined where (x*2 + 2x + 5)/x**2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where (x*2 + 2x + 5)/x**2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where (x*2 + 2x + 5)/x2 = 0 undefined where 1 + 2/x + 5/x2 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 + 2/x + 5/x**2 = 0 undefined where x = 0
8	✓ 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: pass — The solution correctly applies algebraic manipulation to evaluate the limit at infinity. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Senior review claude-sonnet-5-5, 2026-10-09: fail — Line 4 writes the numerator as x/x, which is 1, instead of (x+2)/x = 1+2/x. The later lines inherit this error. The final answer 1 is correct only by coincidence, because the dropped 2/x term tends to 0. The solution should carry (1+2/x) in the numerator through to the limit step.

gpt-oss:20b: uphold — Dividing the numerator x+2 by x gives 1+2/x, not x/x = 1. Line 4 therefore changes the function: it is not equal to line 3 for finite x, and only the limit values happen to coincide, which SymPy's check on the Limit objects could not distinguish.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies algebraic manipulation to evaluate the limit at infinity. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
claude-sonnet-5-5: fail (error) 2026-10-09 — Line 4 writes the numerator as x/x, which is 1, instead of (x+2)/x = 1+2/x. The later lines inherit this error. The final answer 1 is correct only by coincidence, because the dropped 2/x term tends to 0. The solution should carry (1+2/x) in the numerator through to the limit step.
gpt-oss:20b: fail (error) 2026-10-09 — Step 4 incorrectly rewrites the expression: it replaces the numerator (x+2) with x/x and the denominator with sqrt((x**2+2*x+5)/x**2). This is not a valid single-rule application and changes the expression incorrectly.
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies algebraic manipulation and limit laws to evaluate the limit. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.

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 2026-10-09 with SymPy 1.14.0.