Limit of \( \displaystyle \frac{\sqrt{\left(2 x + 1\right)^{2} + 1}}{2 x + 1} \) as \( x \to \infty \)
Problem 1.412 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{\sqrt{\left(2 x + 1\right)^{2} + 1}}{2 x + 1} \).
- \[ \lim_{x \to \infty}\left(\frac{\sqrt{\left(2 x + 1\right)^{2} + 1}}{2 x + 1}\right) \]limitStart with the limit of the given function.✓ Proved
- \[ = \lim_{x \to \infty} \sqrt{1 + \frac{1}{\left(2 x + 1\right)^{2}}} \]algebra simplifyDivide the numerator and denominator by (2*x + 1). Simplify the expression inside the square root.✓ Proved
- \[ = \lim_{x \to \infty} 1 \]limit-lawEvaluate the limit of the term 1/(2*x + 1)**2.✓ Proved
- \[ = 1 \]limitCalculate the final value.✓ 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. The reviewers disagree about how one step is explained; every verdict is in 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 2*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 1 = 0 |
| 5 | ✓ 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 (misleading) — The rewrite in step 2 replaces √((2x+1)^2+1)/(2x+1) with √((2x+1)^2/(2x+1)^2+1/(2x+1)^2). This implicitly assumes (2x+1)>0 so that √((2x+1)^2)/(2x+1)=1, but for large negative x the denominator is negative and the limit would be –1. The step therefore omits the necessary absolute‑value consideration, leading to a misleading conclusion.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-08 — The rewrite in step 2 replaces √((2x+1)^2+1)/(2x+1) with √((2x+1)^2/(2x+1)^2+1/(2x+1)^2). This implicitly assumes (2x+1)>0 so that √((2x+1)^2)/(2x+1)=1, but for large negative x the denominator is negative and the limit would be –1. The step therefore omits the necessary absolute‑value consideration, leading to a misleading conclusion.qwen3.6:27b-mlx: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies algebraic manipulation to simplify the expression before evaluating the limit. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.