Limit of \( \displaystyle \frac{\sqrt{\left(x + 1\right)^{2} + 1}}{x + 1} \) as \( x \to \infty \)
Problem 1.318 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{\sqrt{\left(x + 1\right)^{2} + 1}}{x + 1} \).
- \[ \lim_{x \to \infty}\left(\frac{\sqrt{\left(x + 1\right)^{2} + 1}}{x + 1}\right) \]limit algebraStart with the original limit expression. Rewrite the denominator to facilitate division by x.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{\sqrt{\left(x + 1\right)^{2} + 1}}{x \left(1 + \frac{1}{x}\right)}\right) \]simplifySimplify the term (x+1)/x.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{\sqrt{\frac{\left(x + 1\right)^{2}}{x^{2}} + \frac{1}{x^{2}}}}{1 + \frac{1}{x}}\right) \]algebraDistribute 1/x^2 into the square root by bringing 1/x inside as 1/x^2.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{\sqrt{\left(1 + \frac{1}{x}\right)^{2} + \frac{1}{x^{2}}}}{1 + \frac{1}{x}}\right) \]simplifySimplify the expression inside the square root.✓ Proved
- \[ = \lim_{x \to \infty} 1 \]limitEvaluate the limit by taking the limit of each term as x approaches infinity.✓ Proved
- \[ = 1 \]simplifyCalculate 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 x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where 1 + 1/x = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 + 1/x = 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 + 1/x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 + 1/x = 0 |
| 7 | ✓ 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: passqwen3.6:27b-mlx: fail (error) — Step 2 applies two algebraic transformations simultaneously: it divides the denominator by x and divides the numerator by x (by bringing 1/x inside the square root). This violates the 'one change per step' rule. Additionally, the transition from Step 1 to Step 2 is not a single algebraic manipulation but a restructuring of the entire fraction.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 2 applies two algebraic transformations simultaneously: it divides the denominator by x and divides the numerator by x (by bringing 1/x inside the square root). This violates the 'one change per step' rule. Additionally, the transition from Step 1 to Step 2 is not a single algebraic manipulation but a restructuring of the entire fraction.gpt-oss:20b: fail (style) 2026-10-06 — Step 4 applies two transformations at once: it rewrites sqrt((x+1)^2+1)/x as sqrt(((x+1)^2+1)/x^2) and then expands the numerator inside the square root. Each step should change only one thing.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies algebraic manipulation to normalize the expression and then evaluates the limit. Each step changes one aspect of the expression and uses valid labels from the fixed 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 2026-10-06 with SymPy 1.14.0.