Limit of \( \displaystyle \frac{\sqrt{4 x^{2} + 9} - 3}{4 x^{2}} \) as \( x \to 0 \)
Problem 1.72 · medium
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{4 x^{2} + 9} - 3}{4 x^{2}} \).
- \[ \lim_{x \to 0^+}\left(\frac{\sqrt{4 x^{2} + 9} - 3}{4 x^{2}}\right) \]limit rationalizeStart with the limit of the original function. Multiply the numerator and denominator by the conjugate of the numerator.✓ Proved
- \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{4 x^{2} + 9} + 3} \]algebra simplify cancelExpand the product in the numerator. Simplify the numerator by subtracting 9. Cancel the common factor of 4*x**2 from the numerator and denominator.✓ Proved
- \[ = \frac{1}{6} \]limit simplify simplifyEvaluate the limit by substituting x = 0. Simplify the square root term. Final result.✓ Proved
Answer \( \frac{1}{6} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where sqrt(4*x**2 + 9) + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(4*x**2 + 9) + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(4*x**2 + 9) + 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(4*x**2 + 9) + 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-30
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.