Limit of \( \displaystyle \frac{2 x + 1}{\sqrt{4 x^{2} + 4}} \) as \( x \to \infty \)
Problem 1.2 · medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{2 x + 1}{\sqrt{4 x^{2} + 4}} \).
- \[ \lim_{x \to \infty}\left(\frac{2 x + 1}{\sqrt{4 x^{2} + 4}}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{4 x^{2} + 4}} + \frac{1}{\sqrt{4 x^{2} + 4}}\right) \]limit-lawSplit the numerator across the denominator.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{4 x^{2} + 4}}\right) + \lim_{x \to \infty} \frac{1}{\sqrt{4 x^{2} + 4}} \]limit-lawApply the sum rule for limits.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{x}{\sqrt{1 + \frac{1}{4 x^{2}}} \left|{x}\right|}\right) + \lim_{x \to \infty} \frac{1}{\sqrt{4 x^{2} + 4}} \]algebraFactor out x^2 from the square root.✓ Proved
- \[ = \lim_{x \to \infty} \frac{1}{\sqrt{1 + \frac{1}{4 x^{2}}}} + \lim_{x \to \infty} \frac{1}{\sqrt{4 x^{2} + 4}} \]simplify algebraSimplify the term inside the first limit. Simplify the coefficient.✓ Proved
- \[ = \lim_{x \to \infty} \frac{1}{\sqrt{1 + \frac{1}{4 x^{2}}}} \]limit algebraEvaluate the limit of the second term as x approaches infinity. The second term evaluates to zero.✓ Proved
- \[ = 1 \]limit simplifyEvaluate the limit of the remaining term. Final calculation.✓ 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. 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 4*x**2 + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 4 = 0 undefined where Abs(x) = 0 undefined where x = 0 undefined where 1 + 1/(4*x**2) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 4 = 0 undefined where Abs(x) = 0 undefined where x = 0 undefined where 1 + 1/(4*x**2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 4 = 0 undefined where x = 0 undefined where 1 + 1/(4*x**2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 4 = 0 undefined where x = 0 undefined where 1 + 1/(4*x**2) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 + 1/(4*x**2) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 + 1/(4*x**2) = 0 |
| 10 | ✓ 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: pass — The solution correctly applies limit laws and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Senior review claude-sonnet-5-5, 2026-10-03: fail — Line 4 is false: 4x²+4 = 4x²(1+1/x²), not 4x²(1+1/(4x²)) (that equals 4x²+1), so the rewritten term is 2x/√(4x²+1) rather than 2x/√(4x²+4). SymPy's 'proved' result most likely came from both limits evaluating to 1, not from the expressions being equal. The final answer 1 is still correct, but the line-4 step is an invalid algebraic identity that a student would learn from.
gpt-oss:20b: dismiss — The sign of x is a domain restriction (x→∞ means x>0, so √(4x²)=2x), which is recorded separately and is never a defect; the objection misses the actual flaw in step 4.
Every verdict on record (7)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies limit laws and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-03claude-sonnet-5-5: fail (error) 2026-10-03 — Line 4 is false: 4x²+4 = 4x²(1+1/x²), not 4x²(1+1/(4x²)) (that equals 4x²+1), so the rewritten term is 2x/√(4x²+1) rather than 2x/√(4x²+4). SymPy's 'proved' result most likely came from both limits evaluating to 1, not from the expressions being equal. The final answer 1 is still correct, but the line-4 step is an invalid algebraic identity that a student would learn from.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies limit laws and algebraic simplifications in single-rule steps. The logic is sound and the final answer is correct.gpt-oss:20b: fail (error) 2026-09-29 — Step 4 rewrites √(4x²+4) as 2x√(1+1/(4x²)) without accounting for the absolute value of x in √(4x²)=2|x|. This ignores the sign of x and therefore is not a valid single-rule transformation for all x, leading to an incorrect simplification in subsequent steps.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies limit laws and algebraic simplifications in single-step increments. The logic is sound and the labels are appropriate.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.